Amendments from Version 2

F1000Research

2046-1402

F1000 Research Limited

London, UK

10.12688/f1000research.172480.3

Research Article

Articles

Neutrosophic Extension of the New Odd Weibull-Inverse Weibull Distribution: Theory and Applications

[version 3; peer review: 2 approved with reservations, 1 not approved]

M. Salih

Ahmed

Conceptualization Formal Analysis Investigation Methodology Supervision Writing – Original Draft Preparation 1 khalaf

sara

Conceptualization Formal Analysis Methodology Supervision Writing – Original Draft Preparation 1 N. Abdullah

Kamal

Formal Analysis Methodology Validation Visualization Writing – Review & Editing 1 A. Noori

Nooruldeen

Data Curation Software Supervision Visualization Writing – Review & Editing https://orcid.org/0000-0001-6315-5083 a 2 1mathematics, Tikrit University, Tikrit, Saladin Governorate, 34001, Iraq 2mathematics, University of Fallujah, Al-Fallujah, Al Anbar Governorate, 31002, Iraq

a nooruldeen.a.noori35508@st.tu.edu.iq

No competing interests were disclosed.

23 3 2026

2025

1454

11 3 2026

2026

This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a Neutrosophic extension of the New odd Weibull inverse Weibull (NNOWIW) distribution, aiming to develop a statistical model capable of handling ambiguous or imprecise data. The mathematical formulation of the proposed distribution was derived by combining Neutrosophic logic with the T-X method, specifying the NCDF, the NPDF and the survival and hazard functions. The statistical probability of the distribution was analyzed. To achieve optimal estimation of the distribution parameters, the maximum likelihood (MLE), least square (LSE), and weighted least squares (WLSE) methods were used, and their performance was evaluated using Monte Carlo simulations. The models efficiency was also tested using real battery life data and compared to competing distributions. The results show that the proposed distribution shows competitive and flexible performance in terms of information criteria (AIC, CAIC, BIC, HQIC) and goodness-of-fit tests suggesting its potential usefulness for modeling complex data. However, further validation on larger and more diverse datasets is required to generalize these findings.

Neutrosophic distributions NNOWIW distribution probability weighted moments entropy measures statistical estimation methods

The author(s) declared that no grants were involved in supporting this work.

Revised Amendments from Version 2

This version includes several important revisions. We clarified the rationale behind the selected neutrosophic parameter values, corrected the description of the empirical section to indicate one real-data application only, and revised the simulation study to better align with the size of the real dataset. Additional textual improvements were also made to enhance clarity and consistency across the manuscript.

1. Introduction

Modeling failure and lifespan data is a fundamental topic in applied statistics, given its importance in various fields such as engineering, medicine, economics, and the environment. Among the classical distributions widely used in this field are the Weibull and inverse Weibull distributions, due to their flexibility in representing increasing and decreasing hazard functions. However, the limitations of these models in handling complex data have prompted many researchers to propose more sophisticated and flexible distributions.

One of the most prominent methods that contributed to the emergence of these new distributions is the T-X method, proposed by Alzaatreh et al. in 2013, ¹ which opened the door to generating a large number of general-form distribution families based on basic distributions. This method has led to the development of a wide range of generative families, including: A modified T-X family, ² A new logarithmic family, ³ Odd inverted Topp Leone-H family, ⁴ Shifted exponential-G family, ⁵ Generalized Odd Maxwell family, ⁶ Odd Lomax-G family, ⁷ hybrid Odd exponential- Φ . ⁸

Despite the wide variety of these generative distributional families, most of them assume that the data used are exact and precise. This means they do not account for ambiguity, indeterminacy, or partial information. This represents a clear knowledge gap in the field of modeling real-life data, especially in medical, industrial, environmental fields, where data are often incomplete, derived from expert estimates, or subject to ambiguity. Furthermore, these families, despite their flexibility, have not been used within a Neutrosophic logic framework, which weakens their ability to comprehensively handle uncertain or probable data.

The importance of this research stems from bridging this gap by combining the flexibility of statistical models generated by T-X method with the power of uncertainty representation provided by Neutrosophic logic, to construct an integrated probabilistic model capable of handling both ambiguous. The proposed distribution represents a step toward creating more realistic and accurate statistical tools for analyzing non-deterministic phenomena.

Unlike fuzzy logic, which primarily represents uncertainty by degrees of membership, and Bayesian approaches, which rely on prior assumptions, Neutrosophic logic explicitly incorporates truth (T), falsity (F), and indeterminacy (I) into the modeling framework. This three-component structure allows the proposed NNOWIW distribution to handle interval-valued parameters and contradictory information more effectively, which is particularly useful in real datasets with incomplete or imprecise observations. This distinctive feature provides an additional layer of flexibility and realism compared to traditional approaches, thereby strengthening the motivation for introducing the Neutrosophic extension.

This paper is organized as follows: Section 2 presents the mathematical formulation of the new Neutrosophic inverse Weibull distribution (NNOWIW), including the probability density function (NPDF), the cumulative distribution function (NCDF), and the survival and hazard functions. Section 3 covers the basic statistical properties of the proposed distribution, such as function expansions, the moment generating function MGF, and probability weighted moments PWM, as well as entropy measures (Renyi, Tsallis, Havrda-Charvat). Section 4 reviews parameter estimation method (MLE, LSE, and WLSE), presents the required derivatives of the maximum likelihood functions. Section 5 presents a Monte Carlo simulation study to evaluate the efficiency of the estimation methods according to metrics such as MSE, RMSE, and bias. Section 6 then highlights the practical applications of the proposed distribution by analyzing real battery life data and comparing it with competing distributions using statistical fit measures (AIC, BIC, HQIC) and goodness-of-fit tests. Finally, Section 7 concludes the paper with a summary of the main results and conclusions, indicating some potential avenues for future work.

2. The mathematical formulation of the NNOWIW distribution

The section discusses the mathematical representation of the NNOWIW distribution by presenting its basic components, including the CDF, the hazard function and survival function. Graphs illustrating the behavior of the understanding of the formal properties of the distribution. The CDF of the NNOWIW distribution is represented as follows: Definition 1:

Let the Neutrosophic random variable be represented as X N = d + t I , where t I ∈ [ X L , X U ] . The values X L and X U denoted the lower and upper bounds of the uncertain part of the random variable belonging to the new Neutrosophic odd Weibull family NOWG. This variable consists of a deterministic component d and an indeterminate t I ∈ [ I L , I U ] .When the indeterminacy vanishes X L = X U , the Neutrosophic model reduces to the classical new odd Weibull family. The NPDF and NCDF are characterized by the Neutrosophic shape parameter η N ∈ [ η L , η U ] and ζ N ∈ [ ζ L , ζ U ] , and they take the following general form: F NNOWG ( x N , η N , ζ N , ε ) = 1 − e ( − η N [ − H ( x N , ε ) . log ( 1 − H ( x N , ε ) ) ] ζ N ) (1) f NNOWG ( x N , η N , ζ N , ε ) = η N ζ N h ( x N , ε ) [ H ( x N , ε ) 1 − H ( x N , ε ) − log ( 1 − H ( x N , ε ) ) ] × [ − H ( x N , ε ) . log ( 1 − H ( x N , ε ) ) ] ζ N − 1 e ( − η N [ − H ( x N , ε ) . log ( 1 − H ( x N , ε ) ) ] ζ N ) (2) Where H ( x N , ε ) and h ( x N , ε ) are NCDF and NPDF of baseline distribution with ε parameter. ⁹

Definition 2:

Let the Neutrosophic random variable be represented as X N = d + t I , where t I ∈ [ X L , X U ] . The values X L and X U denote the lower and upper bounds of the uncertain part of the random variable belonging to the Neutrosophic inverse Weibull distribution (NIW). This variable consists of a deterministic component d and an indeterminate t I ∈ [ I L , I U ] .When the indeterminacy vanishes X L = X U , the Neutrosophic model reduces to the classical inverse Weibull distribution. The NPDF and NCDF are characterized by the Neutrosophic shape parameter k N ∈ [ k L , k U ] and n N ∈ [ n L , n U ] , and they take the following general form: H ( x N , k N , n N ) = e − k N x N − n N (3) h ( x N , k N , n N ) = k N n N x N − ( n + 1 ) e − k N x N − n N (4) Where x N , k N , n N > 0 such as x N is Neutrosophic random variable, and k N , n N are neutrosophic shape parameters.

The NCDF of the proposed NNOWIW distribution is derived through appropriate substitution into the general form Equation (3) into (1) as follows: F NNOWIW ( x N , η N , ζ N , k N , n N ) = 1 − e ( − η N [ − e − k N x N − n N . log ( 1 − e − k N x N − n N ) ] ζ N ) (5)

The NPDF of the NNOWIW distribution is derived either by differentiating Equation (5) or by substituting Equations (4) and (3) into Equation (2), as detailed below f NNOWG ( x N , η N , ζ N , k N , n N ) = η N ζ N k N n N x N − ( n + 1 ) e − k N x N − n N [ e − k N x N − n N 1 − e − k N x N − n N − log ( 1 − e − k N x N − n N ) ] × [ − e − k N x N − n N . log ( 1 − e − k N x N − n N ) ] ζ N − 1 e ( − η N [ − e − k N x N − n N . log ( 1 − e − k N x N − n N ) ] ζ N ) (6)

The following expression is used to derive the Neutrosophic survival function, as stated in Ref. 9: S ( x ) = e ( − η N [ − e − k N x N − n N . log ( 1 − e − k N x N − n N ) ] ζ N ) (7)

The calculation of the neutrosophic hazard functions for the NNOWIW distribution is based on the following formula, as presented in the Ref. 10: h ( x ) = η N ζ N k N n N x N − ( n + 1 ) e − k N x N − n N [ e − k N x N − n N 1 − e − k N x N − n N − log ( 1 − e − k N x N − n N ) ] × [ − e − k N x N − n N . log ( 1 − e − k N x N − n N ) ] ζ N − 1 (8)

Figure 1 includes a representation of the NCDF of the NNOWIW distribution, using variable intervals for its Neutrosophic coefficients. Figure 2 includes a representation of the NPDF of the NNOWIW distribution, using variable intervals for its Neutrosophic coefficients. Figure 3 shows the Neutrosophic survival function (NSF) for the NNOWIW distribution, plotted using different values for its Neutrosophic coefficients.

Figure 1. NCDF for NNOWIW distribution. Figure 2. NPDF for NNOWIW distribution. Figure 3. NSF of NNOWIW distribution.

The parameter values were chosen as neutrosophic intervals to represent the degree of uncertainty in the system under consideration. These intervals were designed to cover different value levels for the parameters η , ζ , k , and n , allowing for the representation of multiple patterns of distribution behavior. Lower interval boundaries generate more concentrated distributions, while upper boundaries produce distributions with greater spread or heavier tails. Thus, the graphs are used to highlight the flexibility of the proposed model. Figure 1 shows the behavior of the NCDF, where parameter variations lead to different probability accumulation rates, demonstrating the model's ability to represent varying probability growth rates. Figure 2 shows that the NPDF can take different forms in terms of skewing, peak height, and tail behavior, reflecting the distribution's flexibility in modeling multiple data types. Figure 3 shows that the NSF can take decreasing or progressively changing patterns, confirming the model's ability to represent different failure rate behaviors in reliability and lifetime analysis applications.

3. Statistical properties

This section provides a detailed examination of the statistical properties of the NNOWIW distribution, beginning with the function expansions of both the NCDF and NPDF, followed by the derivation of the quantile function. These expansions facilitate the extraction of several key statistical measures, including the non-central moments, moment generated function, incomplete moment, Lorenzo and Bonferroni curves, probability weighted moments, the characteristic function, and entropy measures.

3.1 NCDF and NPDF expansion

Due to the mathematical complexity of the NCDF and NPDF in Equations (5) and (6), respectively, both functions have been simplified to facilitate the analysis and derivation of the statistical properties of the NNOWIW distribution. This simplification is based on the binomial expansion, exponential function expansion, and logarithmic expansion. Accordingly, the simplified form of the NCDF is obtained as follows: F ( x N ) = 1 − ϑ e − ( j N + 2 i N ζ N ) k N x N − n N Where ϑ = ∑ i N = j N = 0 ∞ ( − 1 ) i N + i N ζ N + j N i N ! η N i N d i N ζ N , j N , and d i N ζ N , j N = j N − 1 ∑ m = 1 j N m ( i N ζ N + 1 ) − j N m + 1 for j N ≥ 0 and d i N ζ N , 0 = 1 (9)

Similarly, the NPDF can be expanded to derive the following form: f ( x N ) = ϒ x N − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + p N ) k N x N − n N − Φ x N − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + z N ) k N x N − n N (10)

Where ϒ = ∑ i N = j N = p N = 0 ∞ ( − 1 ) i N + i N ζ N + ζ N − 1 + j N + p N i N ! η N i N + 1 d i N ζ N + ζ N − 1 , j N ζ N k N n N

And Φ = ∑ i N = j N = Z N = 0 ∞ ( − 1 ) i N + i N ζ N + ζ N − 1 + j N + Z N i N ! η N i N + 1 d i N ζ N + ζ N − 1 , j N ζ N k N n N

As d 1 , z N = z N − 1 ∑ m = 1 j N 2 m − j N m + 1 for z N ≥ 0 and d 1 , 0 = 1

And d i N ζ N , j N = j N − 1 ∑ m = 1 j N m ( i N ζ N + 1 ) − j N m + 1 for j N ≥ 0 and d i N ζ N , 0 = 1

3.2 The quantile function

The quantile function is one of the fundamental tools in analyzing the statistical properties of the NNOWIW distribution. It’s obtained by inverting the NCDF, which enables Monte Carlo simulation and provides deeper insights into distributional characteristics such as skewness, kurtosis, and the median. For the NNOWIW distribution, the quantile function Q ( F ) is derived by solving the equation F ( Q ( F ) ) = b N where b N is a probability value within the open interval (0,1). This function is expressed as Q ( b N ) = F − 1 ( b N ) , and it is derived for the NNOWIW distribution as follows ¹⁰: QF = [ log [ [ − log ( 1 − b N ) η N ] 1 ζ N [ − log ( 1 − b N ) η N ] 1 ζ N + W − 1 [ − log ( 1 − b N ) η N ] 1 ζ N e [ − log ( 1 − b N ) η N ] 1 ζ N ] k N ] − 1 n N (11)

Where W(.) refers to the Lombard function.

Table 1 shows the values of the quantile function at different intervals parameters.

Table 1. Quantile values for selected parameters of the NNOWIW distribution.

s N	( η N , ζ N , k N , n N )
s N	[0.3,1,3], [1.7,2.7], [0.5,1.5], [0.8,1.8]	[0.4,1.4], [1.5,2.5], [0.4,1.4], [0.7,1.7]	[0.5,1.5], [1.6,2.6], [0.8,1.8], [0.6,1.6]	[0.2,1.2], [1.4,2.4], [0.7,1.7], [0.9,1.9]	[0.8,2.8], [1.9,2.9], [0.9,1.9], [0.5,1.5]
0.1	[0.95815, 1.60399]	[0.52946, 1.52535]	[1.38501, 1.84166]	[1.60076, 1.62682]	[1.67147, 2.02330]
0.2	[1.51982, 1.77943]	[0.89207, 1.70852]	[2.06871, 2.37971]	[1.81078, 2.8197]	[2.26463, 2.78811]
0.3	[1.91413, 2.17968]	[1.34554, 1.85107]	[2.24499, 3.60887]	[1.9542, 4.54820]	[2.44921, 4.06494]
0.4	[2.03432, 3.02764]	[1.96309, 1.97972]	[2.40373, 5.24985]	[2.08383, 7.25026]	[2.61333, 5.63795]
0.5	[2.15078, 4.19339]	[2.10565, 2.86448]	[2.55877, 7.58194]	[2.21102, 11.86537]	[2.77175, 7.68571]
0.6	[2.27117, 5.92107]	[2.23713, 4.29272]	[2.72025, 11.15010]	[2.34409, 20.66243]	[2.9348, 10.51989]
0.7	[2.40428, 8.76149]	[2.38404, 6.83650]	[2.90014, 17.21563]	[2.49320, 40.31431]	[2.93482, 14.78851]
0.8	[2.5657, 14.30078]	[2.56443, 12.34952]	[3.12033, 29.52009]	[2.67694, 97.64122]	[3.33080, 22.17841]
0.9	[2.80027, 29.88561]	[2.83047, 30.68424]	[3.44352, 66.19065]	[2.9492, 398.04552]	[3.64266, 39.41633]

3.3 Non-center moments

The non-central moments are fundamental statistics that provide deep insight into the distribution’s key characteristics, including measures of central tendency, variability, skewness, and kurtosis. These moments are essential for both theoretical developments and practical applications, as they help quantify the shape and behavior of the proposed NNOWIW distribution. The r th non-central moment is defined as the expected value of x r , and for the NNOWIW model, it is derived as follows ^{11,
12}: μ r = E ( x N r ) NNOWIW = ∫ − ∞ ∞ x N r f ( x N ) d x N μ r = ϒ ∫ 0 ∞ x N r − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + p N ) k N x N − n N d x N − Φ ∫ 0 ∞ x N r N − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + z N ) k N x N − n N d x N μ r = ϒ I 1 − Φ I 2 where I 1 = ∫ 0 ∞ x N r − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + p N ) k N x N − n N d x N I 2 = ∫ 0 ∞ x N r − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + z N ) k N x N − n N d x N

For I 1 , let y = ( 2 i N ζ N + 2 ζ N + j N + p N ) k N x N − n N I 2 = ∫ 0 ∞ x N r − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + z N ) k N x N − n N d x N x = y 1 n N k N 1 n N ( 2 i N ζ N + 2 ζ N + j N + p N ) 1 n ⟹ d x N = y 1 n N − 1 n N k N 1 n N ( 2 i N ζ N + 2 ζ N + j N + p N ) 1 n dy

Then I 1 = ∫ 0 ∞ ( y 1 n N k N 1 n N ( 2 i N ζ N + 2 ζ N + j N + p N ) 1 n ) r − ( n N + 1 ) e − y y 1 n N − 1 n N k N 1 n N ( 2 i N ζ N + 2 ζ N + j N + p N ) 1 n dy

By simplify I 1 to get a final form as follows: I 1 = 1 n N k N r − n N n N ( 2 i N ζ N + 2 ζ N + j N + p N ) r − n N n N ∫ 0 ∞ y r − n N n N − 1 e − y dy I 1 = Γ ( r − n N n N ) n N k N r − n N n N ( 2 i N ζ N + 2 ζ N + j N + p N ) r − n N n N

By same way for I 2 to get a final form: I 2 = Γ ( r − n N n N ) n N k N r − n N n N ( 2 i N ζ N + 2 ζ N + j N + z N ) r − n N n N

To get: μ r = Γ ( r − n N n N ) n N k N r − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) r − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) r − n N n N ] (12)

The first four moments are found by substituting the value of r and as follows μ 1 = Γ ( 1 − n N n N ) n N k N 1 − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) 1 − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) 1 − n N n N ] (13) μ 2 = Γ ( 2 − n N n N ) n N k N 2 − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) 2 − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) 2 − n N n N ] (14) μ 3 = Γ ( 3 − n N n N ) n N k N 3 − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) 3 − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) 3 − n N n N ] (15) μ 4 = Γ ( 4 − n N n N ) n N k N 4 − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) 4 − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) 4 − n N n N ] (16)

The skewness and kurtosis of the NNOWIW distribution are calculated respectively as follows ¹³: SK NNOWIW = Γ ( 3 − n N n N ) n N k N 3 − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) 3 − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) 3 − n N n N ] ( Γ ( 2 − n N n N ) n N k N 2 − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) 2 − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) 2 − n N n N ] ) 3 2 (17) KU NNOWIW = Γ ( 3 − n N n N ) n N k N 3 − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) 3 − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) 3 − n N n N ] ( Γ ( 2 − n N n N ) n N k N 2 − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) 2 − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) 2 − n N n N ] ) 2 − 3 (18)

Table 2 shows the values of first 4 moments, variance, skewness, and kurtoses at different intervals parameters.

Table 2. Intervals of selected moments of the NNOWIW distribution.

η N	ζ N	k N	n N	μ ` 1 N	μ ` 2 N	μ ` 3 N	μ ` 4 N	σ N 2	S N	K N
[0.6, 1.6]	[1.4, 2.4]	[0.5, 1.5]	[0.1,1.1]	[0.00145, 0.03546]	[0.00132, 0.01849]	[0.00121, 0.01248]	[0.00112, 0.00942]	[0.00132, 0.01723]	[4.96541, 25.2383]	[27.55528, 640.9644]
		[0.5, 1.5]	[0.2,1.2]	[0.00146, 0.06442]	[0.00134, 0.03546]	[0.00123, 0.02432]	[0.00114, 0.01849]	[0.00134, 0.03130]	[3.64311, 25.19755]	[14.70541, 638.3678]
		[0.6, 1.6]	[0.3,1.3]	[0.00089, 0.06975]	[0.00082, 0.04188]	[0.00076, 0.02967]	[0.00071, 0.02291]	[0.00082, 0.03701]	[3.46154, 32.46292]	[13.06408, 1058.45]
		[0.6, 1.6]	[0.4,1.4]	[0.00089, 0.08294]	[0.00083, 0.05250]	[0.00077, 0.03799]	[0.00073, 0.02967]	[0.00083, 0.04562]	[3.1588, 32.4317]	[10.76472, 1055.875]
	[1.8, 2.8]	[0.7, 1.7]	[0.5,1.5]	[0.00015, 0.05242]	[0.00014, 0.03757]	[0.00013, 0.02905]	[0.00012, 0.02360]	[0.00014, 0.03483]	[3.98889, 80.67237]	[16.71791, 6524.261]
		[0.7, 1.7]	[0.6,1.6]	[0.00015, 0.05599]	[0.00014, 0.04156]	[0.00013, 0.03279]	[0.00013, 0.02699]	[0.00014, 0.03843]	[3.87069, 80.63411]	[15.62387, 6516.331]
		[0.9, 1.9	[0.7,1.7]	[4.57e-05, 0.027091]	[4.36e-05, 0.02143]	[4.17e-05, 0.01764]	[4e-05, 0.01494]	[4.36e-05, 0.02070]	[5.62074, 144.8847]	[32.52101, 21027.26]
		[0.9, 1.9	[0.8,1.8]	[4.58e-05, 0.027992]	[4.38e-05, 0.02263]	[4.2e-05, 0.01890]	[4.03e-05, 0.01618]	[4.38e-05, 0.02185]	[5.55200, 144.8432]	[31.59565, 21011.73]

3.4 Moment generating function

The MGF is an important mathematical tool in statistical and probability. It provides an effective means of deriving properties of probability distributions, such as mean, variance, skewness, and flatness. This function helps uniquely characterize and distinguish and is widely used in theoretical analysis and statistical modeling because it simplifies the process of calculating the different moment of distributions. M x ( t ) = E ( e tx ) = ∫ − ∞ ∞ e tx f ( x N ) dx

After including the moments values derived from our proposed model, we arrived at the following result ¹⁴: M x ( t ) = ∑ v N = 0 ∞ t v N v N ! [ Γ ( r − n N n N ) n N k N r − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) r − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) r − n N n N ] ] (19)

3.5 Incomplete moments

Incomplete moments represent a natural extension of the concept of traditional statistical moments, as they are calculated over a specific portion of the distribution domain rather than the entire domain. Their importance lies in their ability to characterize the behavior of a distribution within specific ranges, which making them a powerful tool for analyzing of truncated or skewed data, calculating measures such as mean deviation, studying indices of dispersion and asymmetry, and analyzing inequality curves such as the Lorenz and Bonferroni curves. They also play a key role in economic applications, risk studies, and insurance. The rth incomplete moment of the NNOWIW is mathematically defined as ^{15,
16}: M r ( y ) = ∫ − ∞ y x N r f ( x N ) dx M n ( y ) = ∫ 0 y x r ( ϒ x N − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + p N ) k N x N − n N − Φ x N − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + z N ) k N x N − n N ) d x N M n ( y ) = ϒ I 1 − Φ I 2 where I 1 = ∫ 0 y x N r − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + p N ) k N x N − n N dx

and I 2 = ∫ 0 y x N r − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + z N ) k N x N − n N dx

For

I 1 , let t = ( 2 i N ζ N + 2 ζ N + j N + p N ) k N x N − n N x = t 1 n N k N 1 n N ( 2 i N ζ N + 2 ζ N + j N + p N ) 1 n N , when x N = 0 ⟹ t = 0 and if x N = y ⟹ t = ( 2 i N ζ N + 2 ζ N + j N + p N ) k N y − n N ⟹ dy = t 1 n N − 1 n N k N 1 n ( 2 i N ζ N + 2 ζ N + j N + p N ) 1 n N dt I 1 = ∫ 0 ( 2 i N ζ N + 2 ζ N + j N + p N ) k N y − n N ( t 1 n N k N 1 n N ( 2 i N ζ N + 2 ζ N + j N + p N ) 1 n N ) r − ( n N + 1 ) e − t t 1 n N − 1 n N k N 1 n ( 2 i N ζ N + 2 ζ N + j N + p N ) 1 n N dt I 1 = Γ ( ( r − n N n N ) , ( 2 i N ζ N + 2 ζ N + j N + p N ) k N y − n N ) n N k N r − n N n N ( 2 i N ζ N + 2 ζ N + j N + p N ) r − n N n N

Following a similar procedure for I 2 , we arrive at the final form I 2 = Γ ( ( r − n N n N ) , ( 2 i N ζ N + 2 ζ N + j N + z N ) k N y − n N ) n N k N r − n N n N ( 2 i N ζ N + 2 ζ N + j N + z N ) r − n N n N M n ( y ) = ϒ Γ ( ( r − n N n N ) , ( 2 i N ζ N + 2 ζ N + j N + p N ) k N y − n N ) n N k N r − n N n N ( 2 i N ζ N + 2 ζ N + j N + p N ) r − n N n N − Φ Γ ( ( r − n N n N ) , ( 2 i N ζ N + 2 ζ N + j N + z N ) k N y − n N ) n N k N r − n N n N ( 2 i N ζ N + 2 ζ N + j N + z N ) r − n N n N (20)

3.6 Lorenz and Bonferroni curves

The Lorenz curve is a traditional graphical tool used to represent the cumulative distribution of income or wealth and compare this distribution to an ideal line of equality, helping to measure the degree of inequality. The Bonferroni curve an extension and improvement of the Lorenz curve, provides a more accurate analysis by incorporation additional measures of inequality, making it more sensitive to changes in lower-income groups. Thus, the Bonferroni curve provides deeper insights than the Lorenz curve, especially when studying economic inequality. The Lorenz ( L F ( y ) ) and Bonferroni ( B F ( y ) ) curves are mathematically defined by ^{17,
18}: L F ( y ) = 1 μ ∫ 0 y x N f ( x N ) d , B F ( y ) = L F ( y ) F ( x N )

Accordingly, the Lorenz curve L F ( y ) and the Bonferroni curve B F ( y ) are derived, respectively: L F ( y ) = ϒ μ n N ( ( 2 i N ζ N + 2 ζ N + j N + p N ) k N ) 1 n N − 1 Τ ( ( 2 i N ζ N + 2 ζ N + j N + p N ) k N y − n N ) − Φ μ n N ( ( 2 i N ζ N + 2 ζ N + j N + z N ) k N ) 1 n N − 1 Τ ( 1 − 1 n N , ( 2 i N ζ N + 2 ζ N + j N + z N ) k N y − n N ) (21) B F ( y ) = [ ϒ μ n N ( ( 2 i N ζ N + 2 ζ N + j N + p N ) k N ) 1 n N − 1 Τ ( ( 2 i N ζ N + 2 ζ N + j N + p N ) k N y − n N ) − Φ μ n N ( ( 2 i N ζ N + 2 ζ N + j N + z N ) k N ) 1 n N − 1 Τ ( 1 − 1 n N , ( 2 i N ζ N + 2 ζ N + j N + z N ) k N y − n N ) ] 1 − e ( − η N [ − e − k N x N − n N . log ( 1 − e − k N x N − n N ) ] ζ N ) (22)

3.7 Probability-weighted moments

If random variable X follows the NNOWIW distribution, the Probability Weighted Moments (PWM) can be calculated using the following formula ¹⁹: E ( X p ( F ( x ) ) q ) = ∫ 0 ∞ x N p f ( x N ) ( F ( x N ) ) q d x N p = 1 , 2 , 3 , … , q = 0 , 1 , 2 , …

By using the NCDF and NPDF into the expression f ( x N ) ( F ( x N ) ) q , a series expansion can be used to simplify the algebraic formulation, resulting in the following linear representation ¹⁹: E ( X p ( F ( x ) ) q ) = Γ ( r − n N n N ) n N k N r − n N n N [ Ω ( j N ζ N + ζ N + s _ N + 1 ) r − n N n N − Ψ ( j N ζ N + ζ n + r n ) r − n n ] (23)

Where Ω = ∑ i N = j N = p N = s N = 0 ∞ ( − 1 ) i N + 2 j N + ζ N − 1 + p N + s N ( η N + i N η N ) j ! ( q i N ) d j N ζ N + ζ N − 1 , p N η N ζ N k N n N Ψ = ∑ i N = j N = p N = r N = 0 ∞ ( − 1 ) i N + 2 j N + ζ N − 1 + p N + R N ( η N + i N η N ) j ! d j N ζ N + ζ N − 1 , p N η N ζ N k N n N

3.8 Characteristic function

The characteristic function is a fundamental tool in probability and statistics, used to accurately describe the probability distribution of a random variable. It is defined as the expected value of the exponential complex number e itx , where i is the imaginary unit. This function is distinguished by its ability to derive moments and understand the behavior of a sum of random variables, in addition to its persistence even in cases where the moment generating function (MGF) is not present. The Characteristic function can be calculated using the following formula ¹⁸: Q x ( t ) = E ( e itx ) = ∫ 0 ∞ e itx f ( x N ) dx

Using exponential expansion for above equation, to get a form: Q x ( t ) = E ( e itx ) = ∑ q N = 0 ∞ ( it ) q N q N ! [ μ r ] Q x ( t ) = ∑ q N = 0 ∞ ( it ) q N q N ! [ Γ ( r − n N n N ) n N k N r − n N n N [ ϒ ( 2 i N ζ N + 2 ζ N + j N + p N ) r − n N n N − Φ ( 2 i N ζ N + 2 ζ N + j N + z N ) r − n N n N ] ] (24)

3.9 Entropy measures

Entropy is a fundamental concept in information theory and statistics, used to measure the level of uncertainty or randomness in a probability distribution. In the context of continuous lifetime distributions, entropy provides deep insights into the internal properties and variance of the data. This paper relies on three different entropy measures to characterize the proposed distribution. The Rényi entropy measure for NNOWIW distribution, denoted by I R ( c ) , is derived using chain expansions, yielding the following expression ^{20,
21}: I R ( c ) = 1 1 − c log ∫ 0 ∞ f ( x N ) c dx I R ( c ) = 1 1 − c log [ ∫ 0 ∞ ( ϒ x N − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + p N ) k N x N − n N − Φ x N − ( n N + 1 ) e − ( 2 i N ζ N + 2 ζ N + j N + z N ) k N x N − n N ) c dx ]

Using the Binomial Series I R ( c ) = 1 1 − c log [ ∫ 0 ∞ ∑ s N = 0 c ( − 1 ) s N ( c s N ) ϒ c − s N Φ s N x − c ( n N + 1 ) e − ( 2 i N ζ N c + 2 ζc + j N c + p N c − s N p N + s N z N ) k N x N − n N d x N ]

Let u = ( 2 i N ζ N c + 2 ζc + j N c + p N c − s N p N + s N z N ) k N x N − n N ⟹ x = u 1 n N k 1 n N ( 2 i N ζ N c + 2 ζc + j N c + p N c − s N p N + s N z N ) 1 n N d x N = u 1 n N − 1 n N k N 1 n N ( 2 i N ζ N c + 2 ζc + j N c + p N c − s N p N + s N z N ) 1 n N du I R ( c ) NNOWIW = 1 1 − c log [ ∫ 0 ∞ ∑ s N = 0 c ( − 1 ) s N ( c s N ) ϒ c − s N Φ s _ N ( u 1 n N k 1 n N ( 2 i N ζ N c + 2 ζc + j N c + p N c − s N p N + s N z N ) 1 n N ) − c ( n N + 1 ) ] e − u u 1 n N − 1 n N k N 1 n N ( 2 i N ζ N c + 2 ζc + j N c + p N c − s N p N + s N z N ) 1 n N du

Put ϖ = ∑ s N = 0 c ( − 1 ) s N ( c s N ) ϒ c − s N Φ s _ N 1 k − c ( n N + 1 ) + 1 n N ( 2 i N ζ N c + 2 ζc + j N c + p N c − s N p N + s N z N ) − c ( n N + 1 ) + 1 n N I R ( c ) NNOWIW = 1 1 − c log [ ϖ ∫ 0 ∞ u − c ( n N + 1 ) n N + 1 n N − 1 e − u du ] I R ( c ) NNOWIW = 1 1 − c log [ ϖ . Γ ( 1 − c ( n N + 1 ) n N ) ] (25)

Where ϖ = ∑ s N = 0 c ( − 1 ) s N ( c s N ) ϒ c − s N Φ s _ N 1 k − c ( n N + 1 ) n N + 1 n N ( 2 i N ζ N c + 2 ζc + j N c + p N c − s N p N + s N z N ) − c ( n N + 1 ) n N + 1 n N

The mathematical representation of the Havrda and Charvát entropy is given by ²²: I HC ( c ) = 1 2 1 − c − c ( ( ∫ 0 ∞ f c ( x ) dx ) ) 1 c − 1 ) , c > 0 , c ≠ 1

It is noted that the integration used here is very similar to that used in calculating the Renny entropy. Therefore, the Havrda and Charvat entropies for the NNOWIW distribution can be represented as follows: I HC ( c ) = 1 2 1 − c − c ( ( ϖ . Γ ( 1 − c ( n + 1 ) n ) ) 1 c − 1 ) , c > 0 , c ≠ 1 (26)

The mathematical representation of the Tsallis’s Entropy is given by ²³: T c = 1 1 − c ( 1 − ∫ 0 ∞ f c ( x ) dx ) , c > 0 , c ≠ 1

For the NNOWIW random variable, the Tsallis entropy is given by: T c = 1 1 − c ( 1 − ϖ . Γ ( 1 − c ( n + 1 ) n ) ) , c > 0 , c ≠ 1 (27)

The mathematical representation of the Arimoto Entropy is given by ²⁴: A c = c 1 − c ( ( ∫ − ∞ ∞ f c ( x ) dx ) ) 1 c − 1 ) , c > 0 , c ≠ 1

for the NNOWIW random variable, the Arimoto entropy is given by: A c = c 1 − c ( ( ϖ . Γ ( 1 − c ( n + 1 ) n ) ) 1 c − 1 ) , c > 0 , c ≠ 1 (28)

4. Estimation methods

In this section, we review some of the most prominent statistical estimation methods, including the maximum likelihood method (MLE), the least square method (LSE), and the weighted lest squares method (WLSE). These methods aim to find optimal values for parameters, either by maximizing the likelihood function or minimizing the error functions, which contributes to improving the accuracy of the statistical model and the interpretation of its results.

4.1 Maximum likelihood method

In this paragraph, we review one of the most common estimation methods, the MLE which is used to estimate the parameters η N , ζ N , k N , n N for the NNOWIW distribution. Suppose that X 1 , X 2 , … , X n represents a random sample from the NNOWIW distribution, and that x 1 , x 2 , … , x n represents the observed values of this sample. In this case, the log-likelihood function is given by ^{25–
27}: L ( θ N , x N i ) = ∏ i = 1 m η N ζ N k N n N x N − ( n + 1 ) e − k N x N − n N [ e − k N x N i − n N 1 − e − k N x N i − n N − log ( 1 − e − k N x N i − n N ) ] × [ − e − k N x N i − n N . log ( 1 − e − k N x N i − n N ) ] ζ N − 1 e ( − η N [ − e − k N x N i − n N . log ( 1 − e − k N x N i − n N ) ] ζ N ) we compute the log-likelihood: L = m log ( η N ) + m log ( ζ N ) + m log ( k N ) + m log ( n N ) − ( n N − 1 ) ∑ i = 1 m log ( x N i ) − ∑ i = 1 m k N x N i − n N + ∑ i = 1 m log [ e − k N x N i − n N 1 − e − k N x N i − n N − log ( 1 − e − k N x N i − n N ) ] + ( ζ N − 1 ) ∑ i = 1 m log [ − e − k N x N i − n N . log ( 1 − e − k N x N i − n N ) ] − η N ∑ i = 1 m [ − e − k N x N i − n N . log ( 1 − e − k N x N i − n N ) ] ζ N (29)

4.2 Least squares method

This paragraph discusses the use of the LSE method to estimate the parameters of the NNOWIW distribution. This method relies on finding estimated values for the parameters by minimizing the square error function between the theoretical and experimental values. LSE are defined as the values that minimize the function given in Equation (30), thus ensuring that the difference between the theoretical CDF and its experimental counterpart is minimized. ²⁸ φ ( θ N ) = ∑ i = 1 m [ 1 − e ( − η [ − e − k x − n . log ( 1 − e − k x − n ) ] ζ ) − 1 n + 1 ] 2 (30)

4.3 Weighted least squares method

This paragraph presents WLS E s for the parameters η N , ζ N , k N , n N of the NNOWIW distribution. These estimators are obtained by minimizing the function in Equation (31), ensuring that the differences between the theoretical and experimental values are minimized while giving greater weight to more accurate observations. ²⁹ W ( θ N ) = ∑ i = 1 m ( n + 1 ) 2 ( n + 2 ) i ( n − i + 1 ) [ 1 − e ( − η [ − e − k x − n . log ( 1 − e − k x − n ) ] ζ ) − i n + 1 ] 2 (31)

In the Neutrosophic framework, the parameters η N , ζ N , k N , n N are considered as interval-valued rather than single-point estimates, i.e, θ N ∈ [ θ L , θ U ] . For MLE, the likelihood function is maximized within these bounds, yielding estimates expressed as parameter intervals. Similarly, in LSE and WLSE, the minimization of the error functions is performed while accounting for the interval nature of the parameters. This adjustment allows the estimation methods to explicitly reflect the inherent indeterminacy and partial information in the data, which would not be possible under the classical framework.

5. Monte carlo simulation for NMWIW distribution

Monte Carlo simulation is a powerful statistical tool for studying the performance of parameter estimators in complex models. It relies on generating multiple random samples to accurately estimate statistical indicators. In this section, we apply Monte Carlo simulation to the NNOWIW distribution using three estimations: MLE, LSE, WLSE. The effectiveness of these methods is evaluated using three key metrics: RMSE, Bais, across 1000 iteration for different sample size (20, 50, 100, and 200), where η ̂ N = [2.300000, 2.800000], ζ ̂ N = [1.100000,1.600000], k ̂ N = [1.500000, 2.000000], n ̂ N = [1.800000, 2.300000].

Table 3 provides a summary of the simulation results, showing the three performance measures for each sample size, allowing for a practical assessment of the effectiveness of the estimation methods used.

Table 3. Monte Carlo simulations conducted for the NNOWIW.

N	Est.	MLE
N	Est.	η ̂ N	ζ ̂ N	k ̂ N	n ̂ N
20	Mean	[1.979641,2.808468]	[1.650377, 2.541208]	[1.371018, 1.916618]	[2.275432, 2.894149]
	RMSE	[2.301622, 3.441689]	[2.255750, 3.723777]	[0.339292, 0.339420]	[1.101821, 1.511911]
	Bias	[0.008468, 0.320359]	[0.550377, 0.941208]	[0.083382, 0.128982]	[0.475432, 0.594149]
50	Mean	[2.257186, 2.604057]	[1.417445, 2.078858]	[1.405828, 1.929757]	[2.167621, 2.786597]
	RMSE	[2.210803, 3.886184]	[1.518229, 2.168870]	[0.288427, 0.328623]	[0.891500, 1.243002]
	Bias	[0.042814, 0.195943]	[0.317445, 0.478858]	[0.070243, 0.094172]	[0.367621, 0.486597]
100	Mean	[2.157820, 2.874971]	[1.275036, 1.872960]	[1.419953, 1.962169]	[2.039179, 2.557188]
	RMSE	[1.712591, 1.936241]	[0.844228, 1.324007]	[0.235567, 0.280130]	[0.687170, 0.880511]
	Bias	[0.074971, 0.142180]	[0.175036, 0.272960]	[0.037831, 0.080047]	[0.239179, 0.257188]
200	Mean	[2.161109, 2.868329]	[1.248600, 1.808990]	[1.428393, 1.968765]	[2.016098, 2.529044]
	RMSE	[1.581131, 1.835354]	[0.786150, 1.146443]	[0.221539, 0.255802]	[0.623862, 0.779579]
	Bias	[0.068329, 0.138891]	[0.148600, 0.208990]	[0.031235, 0.071607]	[0.216098, 0.229044]

N	Est.	LSE
N	Est.	η ̂ N	ζ ̂ N	k ̂ N	n ̂ N
20	Mean	[2.846505, 3.427019]	[2.235843, 3.309624]	[1.430160, 1.939704]	[1.959109, 2.613173]
	RMSE	[3.994692, 4.684570]	[3.126627, 5.429703]	[0.355523, 0.367673]	[0.906097, 1.293532]
	Bias	[0.546505, 0.627019]	[1.135843, 1.709624]	[0.060296, 0.069840]	[0.159109, 0.313173]
50	Mean	[2.805941, 3.336907]	[1.771400, 2.693227]	[1.471504, 1.977986]	[2.002227, 2.618416]
	RMSE	[3.582572, 3.895227]	[2.100753, 3.696698]	[0.289369, 0.298498]	[0.875348, 1.201017]
	Bias	[0.505941, 0.536907]	[0.671400, 1.093227]	[0.022014, 0.028496]	[0.202227, 0.318416]
100	Mean	[2.599693, 3.242043]	[1.536499, 2.328507]	[1.468512, 1.984972]	[1.943269, 2.512874]
	RMSE	[2.812160, 2.993824]	[1.316990, 2.407019]	[0.238241, 0.244291]	[0.740164, 1.001402]
	Bias	[0.299693, 0.442043]	[0.436499, 0.728507]	[0.015028, 0.031488]	[0.143269, 0.212874]
200	Mean	[2.443353, 3.197604]	[1.488178, 2.172615]	[1.466170, 1.994343]	[1.965599, 2.518734]
	RMSE	[2.329529, 2.847139]	[1.381767, 2.044499]	[0.222375, 0.230711]	[0.727314, 0.966009]
	Bias	[0.143353, 0.397604]	[0.388178, 0.572615]	[0.005657, 0.033830]	[0.165599, 0.218734]

N	Est.	WLSE
N	Est.	η ̂ N	ζ ̂ N	k ̂ N	n ̂ N
20	Mean	[2.683443, 3.451152]	[1.943156, 2.916858]	[1.418015, 1.951516]	[1.963368, 2.565060]
	RMSE	[4.304954, 5.366513]	[2.273889, 3.774405]	[0.334907, 0.341462]	[0.874110, 1.221558]
	Bias	[0.383443, 0.651152]	[0.843156, 1.316858]	[0.048484, 0.081985]	[0.163368, 0.265060]
50	Mean	[2.383447, 3.104138]	[1.547220, 2.289142]	[1.445669, 1.979226]	[1.997216, 2.583011]
	RMSE	[2.516246, 3.122651]	[1.544994, 2.499629]	[0.246347, 0.255672]	[0.784803, 1.086660]
	Bias	[0.083447, 0.304138]	[0.447220, 0.689142]	[0.020774, 0.054331]	[0.197216, 0.283011]
100	Mean	[2.394649, 3.009759]	[1.381489, 2.032377]	[1.462585, 1.982460]	[1.942155, 2.466895]
	RMSE	[2.028642, 2.044832]	[1.017470, 1.644334]	[0.199602, 0.216089]	[0.660842, 0.850296]
	Bias	[0.094649, 0.209759]	[0.281489, 0.432377]	[0.017540, 0.037415]	[0.142155, 0.166895]
200	Mean	[2.322616, 2.936520]	[1.318156, 1.951916]	[1.471708, 1.986872]	[1.956057, 2.471011]
	RMSE	[1.662044, 1.764831]	[0.922021, 1.546783]	[0.181795, 0.191877]	[0.642346, 0.799665]
	Bias	[0.022616, 0.136520]	[0.218156, 0.351916]	[0.013128, 0.028292]	[0.156057, 0.171011]

The results of Table 3 compares the performance of the MLE, LSE, and WLSE estimation methods based on RMSE and bias at sample sizes N=20, 50, 100, and 200, with parameters having a period of 0.5 for each parameter. At N=20, clear differences emerge between the methods. MLE achieves lower bias and error values for most parameters, while LSE registers the highest RMSE values, particularly for the parameter ζ N , indicating its poor performance at small sample sizes. As the sample size increases to N=50 and N=100, the RMSE and bias values decrease for all methods, with MLE remaining the most stable, followed by WLSE. At N=200, the estimation accuracy improves significantly for all methods due to the increasing sample size. However, MLE remains the best because it achieves the lowest RMSE and bias values for most parameters, while WLSE comes in second, and LSE remains the least efficient. Overall, the results confirm that increasing the sample size gradually improves estimation accuracy, and that MLE is the most efficient of the three methods.

6. Application

To demonstrate the effectiveness of the NNOWIW distribution in achieving an accurate fit to the data, we present a practical example using a real-world data sets. This example aims to illustrate the advantages of the NNOWIW distribution and its fit to the data. Table 4 presents a comparative analysis between the NNOWIW distribution and several other distributions using the validated data.

Table 4. CDF functions for comparative distributions.

Distribution	CDF
Neutrosophic New Odd Weibull Inverse Weibull	1 − e ( − η N [ − e − k N x N − n N . log ( 1 − e − k N x N − n N ) ] ζ N )
Neutrosophic beta inverse Weibull	ρB ( e − k N x N − n N , η N , ζ N )
Neutrosophic Kumaraswamy inverse Weibull	1 − ( 1 − ( e − k N x N − n N ) η N ) ζ N
Neutrosophic Exponeted generalized inverse Weibull	( 1 − ( 1 − e − k N x N − n N ) η N ) ζ N
Neutrosophic log-Gamma inverse Weibull	1 − ρ gamma ( − ζ N . log ( 1 − e − k N x N − n N ) , η N )
Neutrosophic Gompertz inverse Weibull	1 − e η N ζ N { 1 − ( 1 − e − k N x N − n N ) − ζ N }
Neutrosophic inverse Weibull	e − k N x N − n N

These distributions were chosen due to their similar mathematical structure and ability to handle uncertain data. All of these distributions belong to extended or modified families of the Inverse Weibull distribution, making them direct competitors to the NNOWIW distribution. They are characterized by their ability to represent ambiguous or time-interval data, which is the primary goal of the NNOWIW model. Four informative criteria were used ^{30–
35}: AIC = − 2 ∑ j = 1 T log f m ( x j / θ ̂ m ) + 2 k CAIC = AIC + 2 k ( k + 1 ) n − k − 1 HQIC = 2 k ln [ ln ( n ) ] − 2 l ( θ ̂ ) BIC = − 2 l ( θ ̂ ) + k log ( n )

In addition to four statistical measures to assess accuracy ³⁶: K − S = sup n | F n ( x ) − F ( x ) | A = − n − 1 n ∑ i = 1 n ( 2 i − 1 ) H H = [ ln F ( x i ) + ( 1 − ln F ( x n + 1 − i ) ) ] W = 1 12 n + ∑ i = 1 n ( F ( x i ) − 2 i − 1 2 n ) 2 p ‐ value = P ( T ≥ T obs | H 0 )

6.1 Data set

The data represent the lifetime of batteries. The lifetime in 100 hours of 23 batteries is given as. ³⁷

[2.9, 3.99], [5.24, 7.2], [6.56, 9.02], [7.14, 9.82], [11.6, 15.96], [12.14, 16.69], [12.65, 17.4], [13.24, 18.21], [13.67, 18.79], [13.88, 19.09], [15.64, 21.51], [17.05, 23.45], [17.4, 23.93], [17.8, 24.48], [19.01, 26.14], [19.34, 26.59], [23.13, 31.81], [23.34, 32.09], [26.07, 35.84], [30.29, 41.65], [43.97, 60.46], [48.09, 66.13], [73.48, 98.04].

Table 5 shows that the number of observations (23) is sufficient for a preliminary statistical analysis, despite the relatively medium sample size. The mean (mean) ranges between [20.59, 28.19], which represents approximately the expected value for the variable under study. However, this value may be affected by outliers, especially in the presence of skewness. The standard deviation (SD) of [15.93, 21.46] indicates a moderate amount of variance among the values. The skewness coefficient (SK) of [1.77, 1.71] indicates a clear positive skew, with most values concentrated on the lower end of the data, with the tail extending to the higher values. This is a common pattern in life expectancy data or similar data. Kurtness (KU) of [3.32, 4.48] indicates that the distribution is steeper than the normal distribution (which has skewness = 3), meaning there is greater centering around the mean values compared to the normal distribution.

Table 5. Descriptive statistics of the data.

Var	N	Mean	SD	Median	Trimmed	Mad	Min	Max	Range	SK	KU	Se
	23	[20.59, 28.19]	[15.93, 21.46]	[17.05, 23.45]	[18.1, 24.89]	[8.08, 11.1]	[2.9, 3.99]	[73.48, 98.04]	[70.58, 94.05]	[1.77, 1.71]	[3.04, 2.71]	[3.32, 4.48]

Table 6 shows a summary of the criteria for selecting the models used in comparing the distributions, while Table 7 shows the values resulting from the statistical tests. Table 8 includes the estimated values of the parameters of each distribution under study.

Table 6. Distribution evaluation criteria results.

Dist.	-Log	AIC	CAIC	BIC	HQIC
NNOWIW	[88.62533, 95.80281]	[185.2507, 199.605]	[187.4729, 201.827]	[189.7926, 204.147]	[186.393, 200.7479]
NBeIW	[89.6844, 97.2003]	[187.3769, 202.431]	[189.5991, 204.653]	[191.9189, 206.973]	[188.5192, 203.573]
NKuIW	[88.9879, 96.65249]	[185.9769, 201.317]	[188.1991, 203.539]	[190.5189, 205.859]	[187.1192, 202.459]
NEGIW	[90.02595, 97.52312]	[188.0574, 203.060]	[190.2796, 205.282]	[192.5994, 207.602]	[189.1997, 204.203]
NLGamIW	[89.81861, 96.8195]	[187.6456, 201.640]	[189.8679, 203.863]	[192.1876, 206.182]	[188.7879, 202.783]
NGoIW	[89.71942, 96.95312]	[187.4389, 201.906]	[189.6611, 204.128]	[191.9808, 206.448]	[188.5811, 203.048]
NIW	[91.39956, 98.65839]	[186.7991, 201.316]	[187.3991, 201.916]	[189.0701, 203.587]	[187.3703, 201.887]

Table 7. Value of the statistical measures.

Dist.	W	A	K-S	p-value
NNOWIW	[0.06499, 0.06663]	[0.35799, 0.36585]	[0.13708, 0.13841]	[0.71967, 0.73024]
NBeIW	[0.10091, 0.10208]	[0.55900, 0.56697]	[0.17874, 0.19775]	[0.28962, 0.40660]
NKuIW	[0.07621, 0.08998]	[0.42074, 0.49959]	[0.14289, 0.17939]	[0.40218, 0.68340]
NEGIW	[0.11057, 0.11098]	[0.61405, 0.61767]	[0.18435, 0.19613]	[0.29854, 0.36938]
NLGamIW	[7.08964, 7.13061]	[45.57148, 45.64410]	[0.99728, 0.99730]	[4.440892e-16, 4.440892e-16]
NGoIW	[0.10245, 0.10330]	[0.57442, 0.58167]	[0.14507, 0.14701]	[0.64978, 0.66569]
NIW	[0.15295, 0.15333]	[0.85354, 0.85425]	[0.20822, 0.20970]	[0.22945, 0.23637]

Table 8. Estimator value interval for parameters by MLE.

Dist.	k ̂ N	ζ ̂ ̂ N	η ̂ N	ζ ̂ N
NNOWIW	[0.48626, 2.23568]	[2.31023, 2.95356]	[1.57355, 2.67725]	[0.55261, 0.63860]
NBeIW	[4.41993, 4.68428]	[2.82197, 3.33479]	[4.46603, 4.66964]	[0.76757, 0.76997]
NKuIW	[3.96826, 4.49734]	[3.25908, 5.86956]	[4.03131, 4.50760]	[0.71588, 0.82231]
NEGIW	[2.62262, 2.91962]	[4.51696, 4.89758]	[4.69441, 4.90350]	[0.68143, 0.68952]
NLGamIW	[4.79923, 5.41314]	[3.51024, 4.72499]	[4.12389, 4.73166]	[0.79649, 0.92844]
NGoIW	[0.01947, 0.03016]	[0.97378, 1.07021]	[2.40513, 2.43095]	[1.34557, 1.43528]
NIW	[25.97501, 38.82790]	[1.33639, 1.34470]	-----	-----

Table 6 shows the results of the distribution evaluation criteria using -Log, AIC, CAIC, BIC, and HQIC. We note that the NNOWIW distribution has the lowest values in most of these criteria, indicating its superiority and better fit compared to other distributions.

Table 7 shows the results of the statistical goodness-of-fit measures: the Cramer-von-Mises (W) statistic, the Anderson-Darling (A) statistic, and the Kolmogorov-Smirnov (K-S) statistic, in addition to the probability value (p-value) for each distribution.Comparing the values shows that the NNOWIW distribution has the smallest values for W, A, and K-S, with a high p-value (exceeding 0.7), indicating that it is the most suitable model for representing the data. In contrast, other distributions, such as NBeIW and NEGIW, showed higher values for these measures, indicating a poorer fit. The NLGamIW distribution, on the other hand, showed very high values, reflecting its poor fit to the data.

Table 8 displays the estimated intervals for the various model parameters using the maximum likelihood (MLE) method. These parameters include η N , ζ N , k N , n N . It is clear that the NNOWIW distribution has balanced parameter ranges compared to other distributions, reflecting its flexibility and ability to accurately represent data. These results demonstrate that NNOWIW provides a better balance between accuracy and reliability in parameter estimation than competing models.

To further evaluate model adequacy, several goodness-of-fit tests were employed, including the Kolmogorov-Smirnov (K-S), Anderson-Darling (A), and Cramer-von Mises (W) tests. These tests were selected because they capture different aspects of discrepancy between the theoretical and empirical distributions: the K-S test is sensitive to overall deviations, while A and W give more weight to tail behavior. Using multiple goodness-of-fit tests provides a comprehensive and reliable assessment of model performance.

The comparative distributions (NBeIW, NKuIW, NEGIW, NLGanIW, NGoIW, and NIW) were chosen as they represent widely used extensions of the Inverse Weibull family in reliability and lifetime data analysis. Benchmarking the proposed NNOWIW distribution against these flexible alternatives highlights its superior performance in terms of parameter stability and goodness-of-fit, thereby demonstrating its practical applicability to real datasets.

The NNOWIW distribution is compared with the experimental histogram data in Figure 4. The fitted neutrosophic model for NNOWIW demonstrates a close fit to the data, effectively capturing variability and uncertainty. The green shaded area indicates the neutrosophic uncertainty range, while the curves (Distribution 1 and Distribution 2) demonstrate the model’s flexibility in adapting to different parameter sets, in Figure 5: The empirical CDFs are compared with the fitted neutrosophic cumulative functions (NCDFs) for the NNOWIW distribution in Figure 5. Both fitted curves follow the steps of the empirical functions exactly with minimal deviation, indicating that the NNOWIW neutrosophic distribution provides an accurate fit to the data and effectively represents its cumulative behavior.

Figure 4. Neutrosophic NOWIW distribution fitted to histogram data. Figure 5. Empirical and fitted NCDFs of the NNOWIW distribution. 7. Conclusion

The study found that the proposed NNOWIW distribution is an effective and flexible model for representing ambiguous and complex data, competitive and accurate performance compared to several competing distributions according to statistical fit criteria and goodness of fit tests. Estimation methods, particularly the maximum likelihood equation MLE, demonstrated their ability to provide accurate parameter estimates with significant improvement as sample size increased. Monte Carlo simulation tests also showed a reduction in bias, and RMSE, confirming the potential effectiveness of the distribution in statistical analysis of real-world data. A notable strength of the model is its ability to provide interval-based parameter estimates rather than single-point values, which offers a more realistic representation of uncertain or imprecise data. However, the study is limited to data with a medium sample size (23 observations) and unidimensional data, and distributions performance was not tested for multidimensional or multiple failure data.

The scope of the research could be expanded in the future by applying the distribution to larger and more diverse datasets in the engineering, environmental, and medical fields. Further research could involve developing Bayesian versions and robust estimators to improve estimation accuracy in the presence of outliers. In addition, it is recommended to integrate the distribution with hazard models, extend it to the analysis of multiple or sequential failure data, and develop R software packages to facilitate its use in practical applications.

Data availability

The study uses a previously published article: The lifetime in 100 hours of 23 batteries. https://doi.org/10.1145/3711896.3737372 or GitHub - microsoft/BatteryML. Interested readers can directly access the dataset. The lifetime in 100 hours of 23 batteries from the cited article using the link above.

References 1

Alzaatreh

Lee

Famoye

: A new method for generating families of continuous distributions. Metron. 2013;71(1):63–79. 10.1007/s40300-013-0007-y

Aslam

Asghar

Hussain

: A modified TX family of distributions: classical and Bayesian analysis. J. Taibah Univ. Sci. 2020;14(1):254–264. 10.1080/16583655.2020.1732642

Wang

Feng

Zahra

: A New Logarithmic Family of Distributions: Properties and Applications. Comput. Mater. Contin. 2021;66(1):919–929. 10.32604/cmc.2020.012261

Hassan

Al-Omari

Hassan

: The odd inverted Topp Leone–H family of distributions: Estimation and applications. J. Radiat. Res. Appl. Sci. 2022;15(3):365–379. 10.1016/j.jrras.2022.08.006

Eghwerido

Agu

Ibidoja

: The shifted exponential-G family of distributions: Properties and applications. J. Stat. Manag. Syst. 2022;25(1):43–75. 10.1080/09720510.2021.1874130

Ishaq

Panitanarak

Abiodun

: The generalized odd maxwell-kumaraswamy distribution: Its properties and applications. Contemp. Math. 2024;5(1):711–742. 10.37256/cm.5120242888

Noori

Khalaf

Khaleel

: A new generalized family of Odd Lomax-G distributions properties and applications. Adv. Theory Nonlinear Anal. Appl. 2023;7(4):1–16. 10.17762/atnaa.v7.i4.278

Mahdi

Khaleel

Gemeay

: A new hybrid odd exponential-Φ family: Properties and applications. AIP Adv. 2024;14(4):045203. 10.1063/5.0187731

Odeyale

Gulumbe

Umar

: New New Odd Generalized Exponentiated Exponential-G Family of Distributions. UMYU Sci. 2023;2(4):56–64. 10.56919/usci.2324.007

Khaleel

Oguntunde

Al Abbasi

: The Marshall-Olkin Topp Leone-G family of distributions: A family for generalizing probability models. Sci. African. 2020;8:e00470. 10.1016/j.sciaf.2020.e00470

Alanaz

Algamal

: Neutrosophic exponentiated inverse Rayleigh distribution: Properties and Applications. Int. J. Neutrosophic Sci. 2023;21(4):36–42. 10.54216/IJNS.210404

Alanaz

Mustafa

Algamal

: Neutrosophic Lindley distribution with application for Alloying Metal Melting Point. Int. J. Neutrosophic Sci. 2023;21(4):65–71. 10.54216/IJNS.210407

Khalaf

Ibrahim

Noori

: [0,1] Truncated Exponentiated Exponential Burr type X Distribution with Applications. Iraq. J. Sci. 2024;65(8):4428–4440. 10.24996/ijs.2024.65.8.25

Gómez

Santoro

Barranco-Chamorro

: A Family of Truncated Positive Distributions. Mathematics. 2023;11(21):4431. 10.3390/math11214431

Bilal

Mohsin

Aslam

: Weibull‐Exponential Distribution and Its Application in Monitoring Industrial Process. Math. Probl. Eng. 2021;2021(1):1–13. 10.1155/2021/6650237

Akarawak

EEE

Adeyeye

Khaleel

: The inverted Gompertz-Fréchet distribution with applications. Sci. African. 2023;21:e01769. 10.1016/j.sciaf.2023.e01769

Abd El-latif

Almulhim

Noori

: Properties with application to medical data for new inverse Rayleigh distribution utilizing neutrosophic logic. J. Radiat. Res. Appl. Sci. 2025;18(2):101391. 10.1016/j.jrras.2025.101391

Teamah

A-EAM

Elbanna

Gemeay

: FRÉCHET-WEIBULL DISTRIBUTION WITH APPLICATIONS TO EARTHQUAKES DATA SETS. Pakistan J. Stat. 2020;36(2):135–147.

Khalaf

Abdullah

Noori

: The impact of overall intervention model on price of wheat. Iraq. J. Sci. 2024;65(2):853–862. 10.24996/ijs.2024.65.2.22

Rényi

: On measures of entropy and information. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, 20 June–30 July 1960. Oakland, CA, USA: University of California Press;1961; pp.547–561.

Al-Habib

Khaleel

Al-Mofleh

: A new family of truncated nadarajah-haghighi-g properties with real data applications. Tikrit J. Adm. Econ. Sci. 2023;19(2):311–333. 10.25130/tjaes.19.61.2.17

Havrda

Charvát

: Quantification method of classification processes. Concept of structural $ a $-entropy. Kybernetika. 1967; vol.3(1): pp.30–35. Reference Source

Tsallis

: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 1988;52(1):479–487. 10.1007/BF01016429

Arimoto

: Information-theoretical considerations on estimation problems. Inf. Control. 1971;19(3):181–194. 10.1016/S0019-9958(71)90065-9

Henningsen

Toomet

: maxLik: A package for maximum likelihood estimation in R. Comput. Stat. 2011;26(3):443–458. 10.1007/s00180-010-0217-1

Afify

Yousof

Nadarajah

: The beta transmuted-H family for lifetime data. Stat. Interface. 2017;10(3):505–520. 10.4310/SII.2017.v10.n3.a13

Bhatti

Hamedani

Korkmaz

: On Burr III Marshal Olkin family: development, properties, characterizations and applications. J. Stat. Distrib. Appl. 2019;6(1):1–21. 10.1186/s40488-019-0101-7

Swain

Venkatraman

Wilson

: Least-squares estimation of distribution functions in Johnson’s translation system. J. Stat. Comput. Simul. 1988;29(4):271–297. 10.1080/00949658808811068

Jamal

Nasir

Tahir

: The odd Burr-III family of distributions. J. Stat. Appl. Probab. 2017;6(1):105–122. 10.18576/jsap/060109

Akaike

: Factor analysis and AIC. Psychometrika. 1987;52(3):317–332. 10.1007/BF02294359

Cavanaugh

: Unifying the derivations for the Akaike and corrected Akaike information criteria. Stat. Probab. Lett. 1997;33(2):201–208. 10.1016/S0167-7152(96)00128-9

Abd El-latif

Alqasem

Okutu

: A flexible extension of the unit upper truncated Weibull distribution: Statistical analysis with applications on geology, engineering, and radiation Data. J. Radiat. Res. Appl. Sci. 2025;18(2):101434. 10.1016/j.jrras.2025.101434

Cordeiro

Alizadeh

Ramires

: The generalized odd half-Cauchy family of distributions: Properties and applications. Commun. Stat. Methods. 2017;46(11):5685–5705. 10.1080/03610926.2015.1109665

Abid

Abdulrazak

: [0, 1] Truncated Fréchet-Weibull and Fréchet Distributions. Int. J. Res. Ind. Eng. 2018;7(1):106–135. 10.22105/riej.2018.100865.1020

Raya

: A new one-parameter G family of compound distributions: copulas, statistical properties and applications. Stat. Optim. Inf. Comput. 2021;9(4):942–962. 10.19139/soic-2310-5070-1239

Al-Saqal

Hadied

Algamal

: Modeling bladder cancer survival function based on neutrosophic inverse Gompertz distribution. Int. J. Neutrosophic Sci. 2025;25(1):75. 10.54216/IJNS.250106

Zhang

Gui

Zheng

: BatteryML: An open‑source platform for machine learning on battery degradation. Proceedings of the 12th International Conference on Learning Representations (ICLR). GitHub - microsoft/BatteryML;2024.

10.5256/f1000research.197567.r470290

Reviewer response for version 3

Masood

Bushra

1 Referee https://orcid.org/0009-0005-5007-2752 1University of California, Santa Barbara, California, USA

Competing interests: No competing interests were disclosed.

28 3 2026

2026

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation

approve-with-reservations

To account for uncertainty through an indeterminate component, the paper suggests a neutrosophic extension of the Odd Weibull–Inverse Weibull distribution.

Together with simulation research and an actual data application, the authors define the characteristics of the suggested model and use graphical representations to show its behavior. Though several parts of the book need to be clarified and improved, the concept of integrating uncertainty within a neutrosophic framework is intriguing and pertinent.

Although the definition of the neutrosophic component is ambiguous and varied, the presentation is typically intelligible. Specifically, there is confusion since the indeterminate term t _I is presented using distinct notations (e.g., t _I ∈ [XL, XU] and subsequently �� _I ∈ [��, ��]). This component should be defined clearly and consistently by the writers, who should also use consistent notation throughout.

Furthermore, there is room for development in the graphical depiction. The usage of "Line 1" and "Line 2" in Figures 1, 2, and 3 is conceptually inappropriate because the model is interval-based. Instead, the authors should provide parameter intervals and use appropriate labeling to fully explain them in the figure legend.

Additionally, comparisons with current distributions should be included in the application section's fitted PDF and empirical CDF displays. This would make it easier to evaluate the model's performance and show if it is better than conventional options.

Lastly, the conclusion essentially restates the findings and should be improved by emphasizing the key contributions, going over the practical ramifications, and offering potential avenues for future study.

Overall, the paper offers a potential concept; nevertheless, to increase the work's scientific quality, the problems with the description of the neutrosophic component, figure clarity, lack of comparative analysis in the application, and a weak conclusion should be fixed.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Partly

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Statistical modeling, probability distributions, distribution theory, neutrosophic statistics, directional statistics, simulation studies, parameter estimation, goodness-of-fit analysis, machine learning, and time series analysis.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

10.5256/f1000research.197567.r464962

Reviewer response for version 3

Zulkifly

Mohammad Izat Emir

1 Referee https://orcid.org/0000-0002-5311-5886 1Universiti Teknologi Malaysia (UTM), Johor, Malaysia

Competing interests: No competing interests were disclosed.

26 3 2026

2026

recommendation

reject

The paper proposes a neutrosophic extension of the new odd Weibull inverse Weibull family, derives its distributional functions and several theoretical properties, studies parameter estimation via MLE/LSE/WLSE, evaluates estimator performance by Monte Carlo simulation, and illustrates the model with one battery lifetime dataset of 23 interval-valued observations. The paper itself also states that broader validation on larger and more diverse datasets is still needed. The manuscript addresses a familiar genre of statistical distribution paper: it introduces a new model, derives its main functions, studies analytical properties, proposes estimation procedures, runs a simulation study, and compares the model with competing distributions on real data. The stated motivation is that many generated lifetime families assume exact observations, whereas neutrosophic logic is intended to represent ambiguity, indeterminacy, and interval-valued uncertainty more explicitly. The article organizes this work into sections on formulation, properties, estimation, simulation, application, and conclusion. Version 3 also notes that the authors revised the empirical section to clarify that there is only one real-data application, clarified the rationale for parameter values, and revised the simulation to better align with the size of the real dataset.

In principle, that is a reasonable and indexable structure for a methodological paper in applied probability or mathematical statistics. However, in its present form, the manuscript is still only partly satisfactory on all six questions. The main issues are not that the paper lacks an idea, but that it lacks enough clarity, rigor, and reproducibility to make the contribution fully convincing. Many equations and symbols are hard to follow, the notation is inconsistent, the estimation and simulation procedures are not described in enough operational detail for straightforward replication, and the empirical support is thin because it relies on a single small battery dataset. Those limitations do not necessarily invalidate the paper, but they do mean the paper still requires substantial revision before it can be considered scientifically robust.

My overall recommendation would be major revision. The topic is potentially suitable for indexing, but several scientific and presentation issues must be resolved before the article can be considered sound and reproducible.

1. Is the work clearly and accurately presented and does it cite the current literature?

Answer: Partly

The manuscript is presented in a recognizable scholarly structure, and it does cite both foundational and recent references. The introduction situates the work within failure-time modeling, T-X generated families, and neutrosophic logic, and the abstract and section outline are coherent at a high level. The paper also includes recent material up to 2025–2026, so the literature is not obviously outdated.

That said, the presentation is only partly clear. The main weakness is the manuscript’s readability and notation. Across the formulation and properties sections, many equations are difficult to parse, symbols are not always introduced cleanly, and the notation shifts between expressions in a way that makes it hard to verify derivations line by line. There are also noticeable language problems, typographical issues, and local inconsistencies, such as the simulation heading referring to “NMWIW” rather than NNOWIW. These are not merely cosmetic problems: in a theoretical statistics paper, notation and exposition are part of the scientific content because readers need them to check correctness. The authors should perform a full technical rewrite focused on precision and consistency. In particular:

- Add a notation table defining every symbol once and using it consistently throughout.

- Re-typeset and re-check every equation, especially in Sections 2-4, so the notation is readable and internally consistent.

- Ensure every derived quantity is explicitly tied to the preceding definition.

- Revise the English carefully, ideally with help from a fluent scientific editor.

- Remove typographical inconsistencies and section-title errors.

- Strengthen the literature review by distinguishing more clearly what is genuinely new here relative to other neutrosophic extensions and inverse-Weibull-based generalizations.

- Clear, correct notation and readable derivations are required, not optional. If the mathematical objects cannot be followed unambiguously, the paper cannot be reliably assessed.

2. Is the study design appropriate and is the work technically sound?

Answer: Partly

The high-level design is appropriate for this kind of paper. The authors define a new family, derive NCDF/NPDF/survival/hazard functions, study properties such as quantiles, moments, entropy, and related functionals, then move to estimation, simulation, and one application. That overall design is standard and acceptable for a methodological distribution paper.

However, technical soundness is only partly established. The theoretical development is extensive, but the current presentation does not always make it possible to verify whether all derivations are correct. Several formulas are written too compactly, some depend on expansions whose convergence conditions or validity domains are not fully discussed, and the transition from classical to neutrosophic interval-valued parameters is asserted more often than it is rigorously operationalized. The manuscript also claims practical flexibility based largely on figures, summary tables, simulation, and one application, but that is not enough by itself to establish broad technical superiority.

A second issue is that the empirical and validation side is thin. The paper now explicitly says there is only one real-data application, and that application uses just 23 batteries. That is acceptable as an illustration, but weak as evidence for strong claims about general usefulness. The abstract itself recognizes that larger and more diverse datasets are still needed. The authors should strengthen technical soundness as follows:

- Add formal verification steps for key derivations, especially those used later in estimation and simulation.

- State assumptions explicitly, including parameter domains, regularity conditions, and any conditions needed for moment existence, expansion validity, and estimator behavior.

- Explain what the neutrosophic extension changes in practice, beyond replacing scalar parameters with intervals.

- Broaden the validation, ideally with more than one real dataset and with comparisons to stronger benchmark models.

- The paper must make the mathematical derivations verifiable and must justify the neutrosophic construction in a technically explicit way. Without that, the paper remains scientifically fragile.

3. Are sufficient details of methods and analysis provided to allow replication by others?

Answer: Partly

The manuscript provides the general formulas for the proposed model and names the estimation methods used: MLE, LSE, and WLSE. It also reports that a Monte Carlo study was conducted with 1000 iterations at sample sizes 20, 50, 100, and 200, using specified parameter intervals. That gives readers the general outline of the workflow.

But replication requires much more than a conceptual outline. At present, the paper does not provide enough implementation detail for another researcher to reproduce the results reliably. The missing or under-specified elements include:

- how random samples were generated from the proposed distribution in the simulation;

- which optimization algorithm was used for MLE/LSE/WLSE;

- how interval-valued parameters were handled computationally;

- what starting values, bounds, stopping criteria, and convergence checks were used;

- whether failed optimizations occurred and how they were treated;

- what software and code environment were used;

- how goodness-of-fit statistics and p-values were computed in the application.

The authors should add a dedicated Replication and Implementation subsection containing:

- pseudocode or a stepwise algorithm for simulation and estimation;

- the exact random-generation method from the quantile function or another sampler;

- software details, package names, and version numbers;

- optimization details, including initial values and constraints;

- how interval outputs were summarized;

- complete formulas and computational details for all fit criteria and goodness-of-fit tests;

- a public code repository or supplementary code appendix.

- Reproducibility details are required. A methodological paper that cannot be independently rerun is not yet scientifically adequate.

4. If applicable, is the statistical analysis and its interpretation appropriate?

Answer: Partly

This question is applicable. The paper includes estimator comparison by simulation and model fitting to data using information criteria and goodness-of-fit measures. In broad terms, these are reasonable tools for a paper of this type. The authors compare MLE, LSE, and WLSE using RMSE and bias, and they compare the proposed model to competitors using AIC/CAIC/BIC/HQIC and several fit statistics.

The problem is that the statistical analysis is not yet reported with enough depth or caution. The simulation study uses only one parameter setting and a limited set of sample sizes, and the application is based on just one dataset with 23 interval observations. The manuscript then interprets the findings rather strongly, especially the claim that MLE is the “best” and that the proposed model is strongly competitive. Those may be plausible conclusions within the limited examples shown, but they are not yet general conclusions. Also, the paper gives formulas for p-values and test statistics, but the excerpted application details are incomplete, and the manuscript needs to show the actual test outputs transparently, not just summary claims. The authors should improve the statistical analysis by:

- expanding the simulation to include multiple parameter scenarios, not just one;

- reporting Monte Carlo variability more fully, such as standard errors or confidence intervals for performance measures;

- discussing numerical instability or convergence failures, if any;

- including the full application results table, with all fit statistics and p-values clearly presented;

- adding at least one or two more real datasets, preferably from different contexts;

- tempering interpretation so conclusions match the scale of the evidence.

- At minimum, the paper must report the statistical procedures and outputs transparently and avoid overgeneralizing from one small application. Those points are required.

5. Are all the source data underlying the results available to ensure full reproducibility?

Answer: Partly

The manuscript does present the battery lifetime data in the application section, at least in part, and indicates that the real data consist of interval-valued lifetimes for 23 batteries. That is a positive step.

Still, the answer is only partly. Full reproducibility requires more than partial inclusion of the data in the manuscript text. The reader should have access to:

- the complete dataset in machine-readable form;

- the exact formatting used for analysis;

- the code that produced the parameter estimates, fit criteria, goodness-of-fit results, figures, and simulation tables.

At present, the manuscript does not provide a complete reproducibility package. Even if the data are visible in the article, the absence of accompanying code and detailed computational files means another researcher may not be able to reproduce Tables 1-3, the application results, or the figures exactly.

The authors should provide:

- the full battery dataset as supplementary CSV or text;

- all simulation scripts;

- all estimation and plotting scripts;

- a README describing how to reproduce every table and figure;

- ideally, a permanent public repository or journal supplement.

- A complete reproducibility package is strongly recommended, and at minimum the full dataset and enough implementation detail to regenerate the main results are required.

6. Are the conclusions drawn adequately supported by the results?

Answer: Partly

The conclusions are directionally consistent with the work shown. The paper demonstrates that the proposed distribution is mathematically flexible, can produce varied shapes in its functions and plots, and performs competitively under the reported simulation and the single battery-data application. The abstract also appropriately adds a caution that broader validation is still needed.

However, the evidence base is still too narrow to support stronger general claims. One simulation setup and one small dataset do not establish that the model is broadly preferable, generally superior, or widely applicable across reliability problems. The strongest defensible conclusion at this stage is more limited: the model appears promising and worthy of further study, not yet definitively established as a superior practical model. The authors should revise the conclusion so that it:

- distinguishes theoretical contribution from empirical validation;

- states clearly that current empirical evidence is preliminary;

- avoids broad superiority claims unless supported by more applications and stronger benchmarking;

- explains what future work is needed to establish general usefulness.

- The conclusions must be narrowed to what the results actually show. This is required.

The following issues should be treated as mandatory revisions:

1. Mathematical clarity and notation

The equations, symbols, and derivation flow must be made unambiguous and internally consistent. At present, the formulation and properties sections are too difficult to verify reliably.

2. Reproducible estimation and simulation details

The paper must specify exactly how data were simulated, how parameters were estimated, what algorithms and software were used, and how interval-valued parameters were computed in practice.

3. Transparent empirical reporting

The application section must show the full dataset, the full comparative results, and the full goodness-of-fit outputs, including p-values and model-estimation details.

4. Appropriate interpretation of evidence

The claims must be aligned with the actual validation scope. With only one small real dataset, the conclusions should be presented as preliminary.

5. Language and technical editing

The manuscript requires thorough editing for grammar, notation consistency, and typographical errors because these currently interfere with scientific interpretation.

Final overall assessment

This manuscript has a potentially indexable core idea and follows a generally appropriate structure for a distribution theory paper. However, it is not yet strong enough in clarity, verification, reproducibility, and empirical support to merit an unqualified positive assessment. My answers remain Partly for all six questions. With substantial revision especially improved derivational clarity, explicit computational detail, complete reproducibility materials, and more careful interpretation, the article could become scientifically sound. At present, it still requires major revision.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Partly

Are all the source data underlying the results available to ensure full reproducibility?

Partly

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Fuzzy Mathematics, Fuzzy Logic, Geometric Modeling, Computer-Aided Geometric Design (CAGD)

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

10.5256/f1000research.196473.r462130

Reviewer response for version 2

Abid

Salah

1 Referee https://orcid.org/0000-0003-3617-6908 1Mustansiriyah University, Baghdad, Baghdad Governorate, Iraq

Competing interests: No competing interests were disclosed.

5 3 2026

2026

recommendation

approve-with-reservations

Dear editor in chief,

There are still fundamental errors in the manuscript that cannot be ignored:

1. The selection of the default parameters was not based on scientific criteria but was arbitrary. It would be better to choose parameters that reflect different patterns of the distribution so that the recommendations represent a wide range of phenomena.

2. The manuscript mentions two applications based on real data, while in reality, we only find one application based on real data.

3. The sample sizes chosen for the simulation are more than or equal to 50, while the sample size used in the application is 23. This creates a disconnect between the applied and experimental (simulation) aspects, thus preventing the application from utilizing the simulation results.

Professor Salah H Abid (PhD)

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Partly

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

statistical distributions, data analysis, models building

Alsaab

Nooraldeen

Mathematics, Anbar General Directorate of Education, Ministry of Education, Fallujah, Iraq

Competing interests: No competing interests were disclosed.

9 3 2026

Response to the Reviewer

We sincerely thank the Reviewer for the careful reading of the manuscript and for pointing out these important issues. We have revised the manuscript accordingly.

Comment 1: The selection of the default parameters was not based on scientific criteria but was arbitrary. It would be better to choose parameters that reflect different patterns of the distribution so that the recommendations represent a wide range of phenomena.

Response:

Thank you for this important observation. We revised the discussion accompanying Figures 1–3 to clarify the rationale for selecting the neutrosophic parameter intervals. The parameters were not intended as arbitrary values; rather, they were chosen to represent different levels of uncertainty and to generate distinct distributional patterns, including more concentrated shapes, wider spread, and heavier-tail behavior. This clarification has now been added explicitly in the revised manuscript to better justify the selected intervals and to show that they were used to illustrate the flexibility of the proposed model across multiple distributional behaviors.

Comment 2: The manuscript mentions two applications based on real data, while in reality, we only find one application based on real data.

Response:

We appreciate the Editor for identifying this inconsistency. The manuscript has been corrected to state clearly that the paper contains one real-data application only, based on the battery lifetime dataset. The previous wording referring to “two real-world datasets” was inaccurate and has been revised throughout the manuscript to avoid any ambiguity and to ensure full consistency between the text and the actual empirical analysis presented.

Comment 3: The sample sizes chosen for the simulation are more than or equal to 50, while the sample size used in the application is 23. This creates a disconnect between the applied and experimental aspects.

Response:

Thank you for this valuable comment. In response, we revised the simulation study to reduce the gap between the Monte Carlo design and the empirical application. The simulation settings were updated to include small-sample scenarios closer to the real-data application size, so that the behavior of the estimators can be assessed under conditions more comparable to the empirical dataset. We also revised the discussion of the simulation section to emphasize this connection more clearly. This modification improves the coherence between the experimental and applied parts of the manuscript and makes the simulation findings more relevant to the real-data analysis.

10.5256/f1000research.190211.r446473

Reviewer response for version 1

Abid

Salah

1 Referee https://orcid.org/0000-0003-3617-6908 1Mustansiriyah University, Baghdad, Baghdad Governorate, Iraq

Competing interests: No competing interests were disclosed.

10 2 2026

2026

recommendation

approve-with-reservations

The research is well-written. I suggest reconsidering the estimation methods and using only the maximum likelihood method to describe the behavior of model parameters, as the other methods used here are outdated and unnecessary to mention; there are better and more modern methods available.

The p-values of the Cramer-von-Mises (W) and the Anderson-Darling (A) tests are not found in table (7). Those values should be placed in the table.

1. The data that is supposed to be factual, as indicated in the manuscript, is not found in the manuscript itself, nor in the source cited in the manuscript as containing that data. Therefore, I have serious doubts about the practical application and the results obtained in the manuscript, as the data in this case is fictitious and untrue. If this is the case, then the results will be fabricated. Please include the data in the manuscript so that we can review it.

2. The manuscript requires significant revisions before it can be considered to be fully scientifically valid. (Approved with Reservations)

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Partly

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

statistical distributions, data analysis, models building