F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.174089.1 Research Article Articles Modified New [0, 1] - Right Truncated Exponential Rayleigh Distribution Structure and Application

[version 1; peer review: 2 approved, 1 approved with reservations]

 J. Mohammed Maysaa Conceptualization Investigation Methodology Supervision Writing – Review & Editing 1 Mohammed Ali T. Investigation Software Validation https://orcid.org/0000-0003-0235-2352 a 1 N. shalan Rehab Project Administration Visualization Writing – Original Draft Preparation 2 Department of Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Iraq Department of Mathematics, University of Baghdad, College of Science, Baghdad, Iraq ali.t.m@ihcoedu.uobaghdad.edu.iq

No competing interests were disclosed.

 21 1 2026 2026 15 102 5 1 2026 Copyright: © 2026 J. Mohammed M et al. 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. Background

In general, truncated distributions provide a flexible framework that allows for more realistic and accurate representation of constrained data compared to using non-truncated distributions, making them a fundamental element in modern statistical models and the analysis of constrained data.

 Methods

Right – truncated [0,1] exponential Rayleigh distribution ([0,1]-RTERD) for lifetime is presented in this paper. The new distribution contains two parameters, one of scale and another for the shape. Statistical functions of the (RTERD), such as cumulative functions (CDF), probability density (PDF), survival, and hazard functions, are presented in terms of mathematical construction and graphics. Besides, discuss the mathematical and statistical properties of the new distribution, including median, moments of the origin, coefficients of skewness and kurtosis, order statistic, moment generating function, the rėnyi entropy, quantile function.

 Results

the parameters of this life expectancy distribution will be estimate using the maximum likelihood estimation method and compare new continuous model with other continuous distributions using statistical information criterion. Finally, apply the ([0,1]-RTERD) to actual datasets to validate the model’s practicality. The ([0,1]-RTERD) frequently produces a better fit and shows more modeling precision when handling lifespan and reliability data, according to comparisons with traditional models.

 Conclusions

The [0,1]-RTERD provides a new framework for more accurately dealing with data restricted to the interval [0,1], where it is characterized by several important properties, including changing the shape, center, and width of the original distribution as a result of the normalization process, which is reflected in the expected value, standard deviation, and other instantaneous properties.

 Right – truncated [0 1] exponential Rayleigh distribution quantile function Order Statistics Entropy Simulation This research was financially supported by the University of Fallujah, Iraq, through its academic research funding program This research was financially supported by the University of Fallujah, Iraq, through its academic research funding program. The support covered data analysis, computational resources, and publication preparation. The authors gratefully acknowledge the University of Fallujah for providing the facilities and financial assistance that enabled this study. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. 1. Introduction

Truncation methods used in statistical studies, particularly truncated distributions, have received significant attention recently. Truncated distributions have demonstrated their sufficiency and flexibility in data processing. The development of classical distributions has attracted considerable attention from researchers over the past decades. These researchers have contributed to numerous techniques and methods, including generative families, generalization, extension, hybridization, and truncation of distributions. These techniques have contributed to adding highly flexible properties to new distributions, particularly in data modeling. We review some of the most important of these techniques here: the T-X system for generating continuous distributions, introduced by Ref. 1, a new family named Weibull-G was presented by Ref. 2, by relying on the truncation method, a new family of continuous distributions was created by Ref. 3, the Gamma-Weibull-G family distributions developed by Ref. 4, an extended of Weibull-G family of distributions created by Ref. 5, a truncated left and right T-X of generalized families of distributions introduced by Ref. 6, truncated Muth generated (NTM-G) family presented by Ref. 7, and right truncated X-Gamma-G of distributions was presented by Ref. 8. The approach of [0,1] right truncated was applied within different distributions see Refs. 9– 11. In addition, merging distributions has also attracted significant interest from researchers who are interested in generating new distributions. Various methods and techniques have been developed to combine two or more distributions. By using the tail distribution functions, the exponential-Weibull distribution was created by Ref. 12, in keeping with same technique Serial Weibull-Rayleigh distribution has been presented by Ref. 13, a new mixture distribution was come out by mixing three distributions, 14 inverse of exponential-Rayleigh proposed by Ref. 15, generalized exponential-Rayleigh introduced by Ref. 16, and Alpha power exponential Weibull model was developed by Ref. 17, with that if X~ ER( δ , λ ), within δ and λ are both scale parameters, the (CDF) and (PDF) are given as following. F ER ( x ; δ , λ ) = 1 − e − ( δx + λ 2 x 2 ) , x , δ , λ > 0 f ER ( x ; δ , λ ) = ( δ + λx ) e − ( δx + λ 2 x 2 )

In statistics, a truncated distribution is known as a conditional distribution, where the condition for obtaining it lies in determining the domain of the parent distribution. Determining the domain can be either by restricting the range from above (right truncated), or restricting it from below (left truncated), or restricting it from both sides (interval truncated).

Based on the above, we present here a new distribution coming from truncating the exponential Rayleigh distribution. This distribution is a combination of two distributions that are considered special cases of the Weibull distribution used, is highly effective in modeling data, especially lifetime data. In what is coming, we will shed light on the mathematical construction of statistical functions and properties of the distribution. Furthermore, estimating parameters and applying it to real data and a simulated study.

 2. [0,1] – Right truncated exponential-Rayleigh distribution

Recently, as a result of the emergence of data of the types of boundaries, percentages, ratios, and rates within the boundary interval, which can be seen in fields such as engineering, economics, and pathology, which deal with infection and mortality rates, the literature on truncated distributions has been rapidly increasing see Refs. 18– 21. First, to drive the (CDF) and (PDF) of [0,1]-RTERD with general forms that are associated with Eqns (1) and (2) are: G RTER ( x ; δ , λ ) = F ER ( x ; δ , λ ) F ER ( 1 ; δ , λ ) , 0 ≤ x ≤ 1 , δ , λ > 0 g RTER ( x ; δ , λ ) = f ER ( x ; δ , λ ) F ER ( 1 ; δ , λ )

Now, if a random variable X~[0,1]-RTERD ( δ , λ ), the corresponding (CDF) and (PDF) according to Eqns (3) and (4) are: G RTER ( x ; δ , λ ) = 1 − e − ( δx + λ 2 x 2 ) 1 − e − ( δ + λ 2 ) , 0 ≤ x ≤ 1 , δ , λ > 0 g RTER ( x ; δ , λ ) = ( δ + λx ) e − ( δx + λ 2 x 2 ) 1 − e − ( δ + λ 2 )

Considering that G RTER ( x ; δ , λ ) = 1 for x > 1 ,

The survival and hazard functions associated with Eqs (5) and (6), respectively, can be given by the following formulas: S RTER ( x ; δ , λ ) = e − ( δx + λ 2 x 2 ) − e − ( δ + λ 2 ) 1 − e − ( δ + λ 2 ) h RTER ( x ; δ , λ ) = ( δ + λx ) e − ( δx + λ 2 x 2 ) e − ( δx + λ 2 x 2 ) − e − ( δ + λ 2 )

3. Methodology and mathematical properties

This section focuses on examining the shapes and mathematical properties of the [ 0 , 1 ] − RTER D.

3.1 The shapes of [0,1]−RTERD

This section discusses the form of the distribution represented by calculating the limits of pdf and hazard functions as they approach to zero and infinity. lim x → 0 g RTER ( x ; δ , λ ) = lim x → 0 ( ( δ + λ x ) e − ( δx + λ 2 x 2 ) 1 − e − ( δ + λ 2 ) ) = δ 1 − e − ( δ + λ 2 ) lim x → ∞ g RTER ( x ; δ , λ ) = ( 1 − e − ( δ + λ 2 ) ) − 1 lim x → ∞ ( δ + λ x e ( δx + λ 2 x 2 ) ) = ( 1 − e − ( δ + λ 2 ) ) − 1 lim x → ∞ ( λ ( δ + λ x ) e ( δx + λ 2 x 2 ) ) = ( 1 − e − ( δ + λ 2 ) ) − 1 ( λ ∞ ) = 0 lim x → 0 h RTER ( x ; δ , λ ) = lim x → 0 ( ( δ + λ x ) e − ( δx + λ 2 x 2 ) e − ( δx + λ 2 x 2 ) − e − ( δ + λ 2 ) ) = δ 1 − e − ( δ + λ 2 ) lim x → ∞ h RTER ( x ; δ , λ ) = lim x → ∞ ( ( δ + λ x ) e − ( δx + λ 2 x 2 ) e − ( δx + λ 2 x 2 ) − e − ( δ + λ 2 ) ) = lim x → ∞ ( δ + λ x 1 − e δ ( x − 1 ) + λ 2 ( x 2 − 1 ) ) = lim x → ∞ ( δ − ( δ + λ x ) e δ ( x − 1 ) + λ 2 ( x 2 − 1 ) ) = − δ ∞ = 0

In the following, through Figures 1, 2, 3, and 4, we present plots (CDF), (PDF), S ( x), and h ( x) functions with deferent parameters values.

Description of the figure of cdf of [0,1]-RTERD. Description of the figure pdf of [0,1]-RTERD. Description of the figure <italic toggle="yes">S</italic> ( <italic toggle="yes">x</italic>) of [0,1]-RTERD. Description of the figure <italic toggle="yes">h</italic> ( <italic toggle="yes">x</italic>) of [0,1]-RTERD.

In the figure above, the four plots represent the basic statistical properties of proposed distribution under different parameters values.

Figure 1 shows the CDF plot, where all curves start from zero and gradually increase towards one as ( x) increases. The results show that increasing the value of ( δ ) accelerates the accumulation of probability, while increasing (λ) slows it down, reflecting the influence of these two parameters in determining the shape and spread of [0,1]-RTERD.

Figure 2 shows the PDF plot with demonstrates that changing the parameters leads to a clear difference in the shape of the distribution and the position of its peak. The peaks tend towards smaller values of ( x) as ( δ ) increases, indicating a leftward bias in the distribution.

Figure 3 shows the survival function ( S( x)), which gradually decreases as x increases. It is evident that increasing δ decreases the probability of survival, while increasing λ slows the rate of decrease.

Figure 4 shows the hazard function ( H( x)), which exhibits an increasing pattern indicating a rising probability of failure or risk as x increases. It is also evident that higher λ values lead to a higher risk rate at higher x values.

Overall, these results demonstrate the flexibility of [0,1]-RTERD in representing different behaviors of probability functions and its ability to characterize data of varying shapes by considering the influence of the parameters on the shape of [0,1]-RTERD, its centering, and the behavior of the risk function.

 3.2 Moments, skewness and kurtosis

Moments are considered one of the most important statistical properties, as they are the basis for some other properties, such as E ( x ) , var ( x ) , skewness (S.C), and kurtosis (K.C). The moments of [0,1]-RTERD are; M r ′ = E ( x r ) = ∫ 0 1 x r ∙ g RTER ( x ; δ , λ ) dx = 1 1 − e − ( δ + λ 2 ) ∫ 0 1 x r ∙ ( δ + λ x ) e − ( δx + λ 2 x 2 ) dx = 1 1 − e − ( δ + λ 2 ) ∫ 0 1 x r ∙ ( δ + λ x ) e − δx e − λ 2 x 2 dx e − λ 2 x 2 = ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! x 2 i M r ′ = ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! 1 − e − ( δ + λ 2 ) ∫ 0 1 x r + 2 i ∙ ( δ + λ x ) e − δx dx

Let y = δx ⇨ x = y δ ⇨ dx = dy δ M r ′ = ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! 1 − e − ( δ + λ 2 ) ∫ 0 1 x r + 2 i ∙ ( δ + λ x ) e − y dx = ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! 1 − e − ( δ + λ 2 ) ∫ 0 1 y r + 2 i δ r + 2 i + 1 ∙ ( δ + λ δ y ) e − y dy = ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! δ r + 2 i + 1 ( 1 − e − ( δ + λ 2 ) ) ( δ ∫ 0 1 y r + 2 i e − y dy + λ δ ∫ 0 1 y r + 2 i + 1 e − y dy ) M r ′ = 1 δ r + 2 i + 1 ( 1 − e − ( δ + λ 2 ) ) ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! ( δ γ ( r + 2 i + 1 , 1 ) + λ δ γ ( r + 2 i + 2 , 1 ) )

Remark. If a random variable X~[0,1]-RTER( δ , λ ), the following are satisfy. M 1 ′ = E ( x ) = 1 δ r + 2 i + 1 ( 1 − e − ( δ + λ 2 ) ) ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! ( δγ ( 2 i + 2 , 1 ) + λ δ γ ( 2 i + 3 , 1 ) ) M 2 ′ = 1 δ r + 2 i + 1 ( 1 − e − ( δ + λ 2 ) ) ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! ( δγ ( 2 i + 3 , 1 ) + λ δ γ ( 2 i + 4 , 1 ) ) M 3 ′ = 1 δ r + 2 i + 1 ( 1 − e − ( δ + λ 2 ) ) ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! ( δγ ( 2 i + 4 , 1 ) + λ δ γ ( 2 i + 5 , 1 ) ) M 4 ′ = 1 δ r + 2 i + 1 ( 1 − e − ( δ + λ 2 ) ) ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! ( δγ ( 2 i + 5 , 1 ) + λ δ γ ( 2 i + 6 , 1 ) ) var ( x ) = M 2 ′ − ( M 1 ′ ) 2 S.C = M 3 ′ ( M 2 ′ ) 3 2 C.K = M 4 ′ ( M 2 ′ ) 2 − 3

Table 1 shows the moment values when r = 1, 2, 3, and 4, as well as the values of variance, skewness, and kurtosis, where the data values were confined to the interval [0,1], which represents the integral limits for the moments.

1 <sup>st</sup>-4 <sup>th</sup> moments, skewness, kurtosis, and variance values for x∈[0,1].
δ λ M 1 ′ M 2 ′ M 3 ′ M 4 ′ K . C S . C var
1.5 2.5 0.3718 0.2036 0.1331 0.0964 -0.6758 1.4489 0.0654
2.5 1.5 0.3028 0.1491 0.0917 0.0639 -0.1264 1.5924 0.0574
2.75 0.25 0.2945 0.1453 0.0903 0.0637 0.0166 1.6304 0.0586
3.75 1.25 0.2355 0.0980 0.0538 0.0347 0.6079 1.7535 0.0426
0.75 3.25 0.4415 0.2617 0.1788 0.1330 1.3355 -1.0585 0.0669
3.3 Moment-generating function

It is one of the important properties that we obtain and is given by the following formula: M X ( t ) = E ( e xt ) = ∫ 0 1 e xt ∙ g RTER ( x ; δ , λ ) dx = 1 1 − e − ( δ + λ 2 ) ∫ 0 1 e xt ∙ ( δ + λ x ) e − ( δx + λ 2 x 2 ) dx = 1 1 − e − ( δ + λ 2 ) ∫ 0 1 ( δ + λ x ) e − ( δ + t ) x − λ 2 x 2 dx = ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! 1 − e − ( δ + λ 2 ) ∫ 0 1 x 2 i ( δ + λ x ) e − ( δ + t ) x dx

Let y = ( δ + t ) x ⇨ x = y δ + t ⇨ dx = dy δ + t M X ( t ) = ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! ( δ + t ) 2 i + 1 ( 1 − e − ( δ + λ 2 ) ) ∫ 0 1 y 2 i ( δ + λ y δ + t ) e − y dy = ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! ( δ + t ) 2 i + 1 ( 1 − e − ( δ + λ 2 ) ) ( δ ∫ 0 1 y 2 i e − y dy + λ δ + t ∫ 0 1 y 2 i + 1 e − y dy ) M X ( t ) = ∑ i = 0 ∞ ( − 1 ) i λ i 2 i i ! ( δ + t ) 2 i + 1 ( 1 − e − ( δ + λ 2 ) ) ( δγ ( 2 i + 1 , 1 ) + λ δ + t γ ( 2 i + 2 , 1 ) )

3.4 Quantile function

Q ( u ) = G − 1 ( u ) , u ∈ ( 0 , 1 ) 1 − e − ( δx + λ 2 x 2 ) 1 − e − ( δ + λ 2 ) = u e − ( δx + λ 2 x 2 ) = 1 − u ( 1 − e − ( δ + λ 2 ) ) δx + λ 2 x 2 = − ln ( 1 − u ( 1 − e − ( δ + λ 2 ) ) ) λ 2 x 2 + δx + ln ( 1 − u ( 1 − e − ( δ + λ 2 ) ) ) = 0 Q ( u ) = x u = 1 λ ( − δ ± δ 2 − 2 λ ln ( 1 − u ( 1 − e − ( δ + λ 2 ) ) ) )

A spatial case of eqn. (18), the median of [0,1]-RTERD is verified when u = 0.5. Q ( 0.5 ) = x 0.5 = 1 λ ( − δ ± δ 2 − 2 λ ln ( 1 − 0.5 ( 1 − e − ( δ + λ 2 ) ) ) )

3.5 Rėnyi entropy

One of the most important properties that play a fundamental role in extracting information is the randomness measure defined as follows 22 : ₮ RTER ( η ) = 1 1 − η log ( ∫ 0 1 g RTER η ( x ) dx ) = 1 1 − η log ( 1 ( 1 − e − ( δ + λ 2 ) ) η ∫ 0 1 ( δ + λx ) η e − η ( δx + λ 2 x 2 ) dx ) ( δ + λ x ) η = ∑ j = 0 ∞ ( η j ) δ η − j λ j x j e − ηλ 2 x 2 = ∑ p = 0 ∞ ( − 1 ) p η p λ p 2 p p ! x 2 p ₮ RTER ( η ) = 1 1 − η log ( ∑ j ∞ ∑ p ∞ ( η j ) ( − 1 ) p η p λ p 2 p p ! ( 1 − e − ( δ + λ 2 ) ) η ∫ 0 1 x 2 p + j e − ηδx dx )

Let y = ηδx ⇨ x = y ηδ ⇨ dx = dy ηδ ₮ RTER ( η ) = 1 1 − η log ( ∑ j ∞ ∑ p ∞ ( η j ) ( − 1 ) p η p λ p 2 p p ! ( ηδ ) 2 p + j + 1 ( 1 − e − ( δ + λ 2 ) ) η ∫ 0 1 y 2 p + j e − y dy ) ₮ RTER ( η ) = 1 1 − η log ( ∑ j ∞ ∑ p ∞ ( η j ) ( − 1 ) p η p λ p 2 p p ! ( ηδ ) 2 p + j + 1 ( 1 − e − ( δ + λ 2 ) ) η γ ( 2 p + j + 1 , 1 ) )

3.6 Order statistic

Let x 1 : K ≤ x 2 : K ≤ x 3 : K ≤ … ≤ x K : K be the order statistics of a random sample x 1 , x 2 , … , x K of size k from [0,1]-RTERD. The PDF for order statistics is as follows: g P , K ( x ) = K ! ( P − 1 ) ! ( K − P ) ! ( G RTER ( x ; δ , λ ) ) P − 1 ( 1 − G RTER ( x ; δ , λ ) ) K − P g RTER ( x ; δ , λ ) = ₭ ( 1 − e − ( δx + λ 2 x 2 ) 1 − e − ( δ + λ 2 ) ) P − 1 ( e − ( δx + λ 2 x 2 ) − e − ( δ + λ 2 ) 1 − e − ( δ + λ 2 ) ) K − P ( δ + λ x ) e − ( δx + λ 2 x 2 ) 1 − e − ( δ + λ 2 )

Where ₭ = K ! ( P − 1 ) ! ( K − P ) ! g P , K ( x ) = ₭ ( 1 − e − ( δx + λ 2 x 2 ) ) P − 1 ( e − ( δx + λ 2 x 2 ) − e − ( δ + λ 2 ) ) K − P ( ( δ + λx ) e − ( δx + λ 2 x 2 ) ) ( 1 − e − ( δ + λ 2 ) ) K

The minimum order statistics of PDF, when P = 1 , and the maximum value among the order statistics, when P = K , of the [0,1]-RTERD, is given in the following form: g 1 , K ( x ) = ₭ ( e − ( δx + λ 2 x 2 ) − e − ( δ + λ 2 ) ) K − 1 ( ( δ + λx ) e − ( δx + λ 2 x 2 ) ) ( 1 − e − ( δ + λ 2 ) ) K g K , K ( x ) = ₭ ( 1 − e − ( δx + λ 2 x 2 ) ) K − 1 ( ( δ + λx ) e − ( δx + λ 2 x 2 ) ) ( 1 − e − ( δ + λ 2 ) ) K

4. Maximum likelihood estimation

In order to estimate the parameters, MLE applied to estimate the two parameters of [0,1]-RTERD. Let x 1 , x 2 , ⋯ x n be random samples that are distributed as [0,1]-RTERD, then the likelihood function is: L ( x 1 , x 2 , ⋯ , x n ; δ , λ ) = ∏ i = 1 n g RTER ( x i ; δ , λ ) = ∏ i = 1 n ( δ + λ x i ) e − ( δ x i + λ 2 x i 2 ) 1 − e − ( δ + λ 2 ) = ( 1 − e − ( δ + λ 2 ) ) − n e − ∑ i = 1 n ( δ x i + λ 2 x i 2 ) ∏ i = 1 n ( δ + λ x i ) L ̂ = ln ( L ) = − nln ( 1 − e − ( δ + λ 2 ) ) − ∑ i = 1 n ( δ x i + λ 2 x i 2 ) + ∑ i = 1 n ln ( δ + λ x i ) ∂ L ̂ ∂ δ = − n e − ( δ + λ 2 ) 1 − e − ( δ + λ 2 ) − ∑ i = 1 n x i + ∑ i = 1 n 1 δ + λ x i ∂ L ̂ ∂ λ = − n e − ( δ + λ 2 ) 2 ( 1 − e − ( δ + λ 2 ) ) − 1 2 ∑ i = 1 n x i 2 + ∑ i = 1 n x i δ + λ x i

After setting Equations (24) and (25) equal to zero, we obtain nonlinear equations whose solutions are found, representing the estimated parameter values, using MATLAB programming. 23

5. Simulation

Monte-Carlo simulation applied to examine statistical performance of MLE for [0,1]-RTERD. The evaluation centered on two primary accuracy indicators: bias and mean squared error (MSE). To achieve this, random samples were generated based on Equation (18), using different sizes ( n = 20, 30, 50, 100, and 200), and repeated across L = 1000 replications. Parameter estimates were obtained via (MLE), with starting values specified as δ = 0.25, λ = 1.5 and δ = 0.1, λ = 0.5. The simulations were implemented using MATLAB (version 2018a).

Table 2 shows the parameter estimates using (MLE) across different sample sizes (20, 30, 50, 100, and 200). The results proved that the accuracy of the estimates gradually improves with increasing sample size, as estimated values approach the true values of the parameters, and their variance decreases, as evidenced by the decline in the MSE, bias, and the mean values for each parameter. In smaller samples, the estimates were less accurate, while they became more reliable and consistent in larger samples. This confirms that the MLE method gains greater accuracy and stability with increasing sample size.

Average estimate, MSE, and Bias values corresponding to MLE.
Parameter n Mean ( δ ̂ ) Mean ( λ ̂ ) MSE ( δ ̂ ) Bias ( δ ̂ ) MSE ( λ ̂ ) Bias ( λ ̂ )
δ = 0.25 λ = 1.5 20 0.2568 1.3027 0.0314 0.0068 0.4748 -0.1973
30 0.2586 1.3092 0.0261 0.0086 0.4639 -0.1908
50 0.2496 1.3618 0.0191 -0.0004 0.4085 -0.1382
100 0.2386 1.4106 0.0121 -0.0114 0.3456 -0.0894
200 0.2371 1.4207 0.0062 -0.0129 0.2976 -0.0793
δ = 0.1 λ = 0.5 20 0.1438 0.6666 0.0198 0.0438 0.1377 0.1666
30 0.1271 0.6387 0.0126 0.0271 0.1275 0.1387
50 0.1193 0.6126 0.0085 0.0193 0.1228 0.1126
100 0.1058 0.6052 0.0041 0.0058 0.1163 0.1052
200 0.1018 0.5988 0.0024 0.0018 0.1043 0.0988

Figures 5 & 6 reinforce the content discussed above in terms of what can be observed regarding the values of estimators, error, and bias.

(a) Description of MSE and Bias Estimation Performances [0,1]-RTERD of (δ = 0.25, λ = 1.5). (b) Description of MSE and Bias Estimation Performances [0,1]-RTERD of (δ = 0.1, λ = 0.5). (a) Description of the Histogram of estimate values of δ = 0.25, λ = 1.5 of [0,1]-RTERD. (b) Description of the Histogram of estimate values of δ = 0.1, λ = 0.5 of [0,1]-RTERD. 6. Application

Information criteria are tools used to demonstrate a distribution’s flexibility and effectiveness in representing realistic data by comparing it to the performance of other distributions. Some of these criteria, such as the AIC , AIC c , BIC , and HQIC , will be applied to a data set representing The data represents the failure time of 50 components ( 10 3 hours), as it is complete data with a rightward skewed nature. These data were utilized by Arshad MZ et al., 24 and are presented as follows:

(0.213, 0.275, 0.099, 0.388, 0.497, 0.087, 0.073, 0.089, 0.061, 0.086, 0.089, 0.790, 0.118, 0.075, 0.297, 0.119, 0.299, 0.315, 0.403, 0.192, 0.308, 0.191, 0.909, 0.168, 0.135, 0.314, 0.117, 0.120, 0.183, 0.714, 0.143, 0.374, 0.859, 0.084, 0.089, 0.215, 0.297, 0.992, 0.330, 0.817, 0.298, 0.258, 0.123, 0.815, 0.088, 0.102, 0.203, 0.199, 0.185, 0.257).

Table 3 shows the values of the information criteria and the performance of [0,1]-RTERD compared to the performance of the other models RTED, RTWD, RTRD, and ERD in representing the data above. It is noticeable that the distribution is capable and effective, as shown in the included results. In addition, the [0,1]-RTERD was compared in terms of representing the pdf’s and cdf’s functions for turbochargers data through Figure 7.

Negative log (L) and information criteria.
Dist. -log(L) AIC BIC AICc HIQC
RTER 12.558 29.103 32.928 29.359 30.559
RTE 17.831 39.661 43.485 39.916 41.117
RTW 17.161 39.322 44.102 39.698 41.142
RTR 21.808 43.716 43.812 43.718 43.752
ER 14.818 33.635 37.459 33.890 35.091
Description of turbochargers data in term of empirical histogram and empirical cumulative [0,1]-RTERD.

Figure 7 show a comparison between the pdf.s and cdf.s for several interpolated [0,1] distributions: RTER, ER, RTE, RTW, and RTR distributions, along with the experimental data.

Figure 7(a) it is evident that [[0,1]-RTER (solid red line) best matches the overall pattern of the sample data, particularly in the initial region of small x-values, where its curve closely resembles the experimental density pattern. The other distributions show relative deviations; RTR tends to overestimate small values, while ER and RTE provide less accurate estimates in the tail regions.

Figure 7(b) highlights degree of agreement between the estimated cumulative functions and the experimental function. We observe that the RTER curve best represents data behavior across the entire interval [0,1], outperforming the other distributions, which either exceed the empirical function or deviate from it at the middle and upper values of ( x).

Overall, the results confirm that the proposed RTER distribution exhibits greater flexibility and better representation of truncated data compared to the other distributions, reflecting its efficiency in modeling real data within the finite interval [0,1].

 Conclusion

The proposed [0,1]-RTERD offers a flexible generalization of the classical exponential and Rayleigh distributions. This enhancement allows the model to accommodate various shapes of hazard functions, improving its suitability for modeling lifetime and reliability data. The properties of the distribution were systematically derived, and parameter estimation was performed using MLE method. To assess the model’s goodness-of-fit, several information-based criteria were employed, including AIC, AIC c , BIC, and HQIC. The results indicate that the RTERD outperforms traditional models, demonstrating a superior fit to real-world data and highlighting its potential as a robust tool in applied reliability and survival analysis.

 Ethics and consent

Since the study did not include sensitive or personal data or human subjects, there is no ethical or approval requirement for this study.

 Data availability Underlying data

Zenodo: Real experimental data supporting this study. https://doi.org/10.5281/zenodo.18071193 or https://github.com/alitm-afk/Application-of-Failure-time-of-50- components. 24, 25

The project contains the following underlying data:

[Application-of-Failure-time-of-50-components-main.zip] (The Failure Time of 50 Components (10 3 hours) Information criteria are tools used to demonstrate a distribution’s flexibility and effectiveness in representing realistic data by comparing it to the performance of other distributions. Some of these criteria, such as the AIC, AIC_c, BIC, and HQIC, will be applied to a data set representing, the data represents the failure time of 50 components (10 3 hours), as it is complete data with a rightward skewed nature).

(contains the raw experimental data used to generate Figure 7 and the results reported in Table 3).

 Extended data

Zenodo: Simulation and numerical data used in this study. https://doi.org/10.5281/zenodo.18119198 26

This project contains the following extended data:

Figures(1,2,3,4)_dataset.xlsx (data used to generate Figures 1–4).

simulation_data.xlsx (simulation results used to generate figures 5 and 6 and results used in Table 2).

moments, skewness, kurtosis, and variance.docx (descriptive statistics and calculations associated with the simulation data and results used in Table 1).

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

 References Alzaatreh A Lee C Famoye F : A new method for generating families of continuous distributions. Metron. 2013;71(1):63–79. 10.1007/s40300-013-0007-y Tahir MH Zubair M Mansoor M : A new Weibull-G family of distributions. Hacettepe J. Math. Stat. 2016;45(2):1–47. 10.1186/s40488-015-0036-6 Mahdavi A Oliveira Silva G : A method to expand family of continuous distributions based on truncated distributions. J. Stat. Res. Iran JSRI. 2017;13(2):231–247. 10.18869/acadpub.jsri.13.2.231 Oluyede B Pu S Makubate B : The gamma-Weibull-G family of distributions with applications. Austrian. J. Stat. 2018;47(1):45–76. 10.17713/ajs.v47i1.155 Korkmaz MÇ : A new family of the continuous distributions: the extended Weibull-G family. Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat. 2018;68(1):248–270. 10.31801/cfsuasmas.451602 Alzaatreh A Aljarrah MA Smithson M : Truncated family of distributions with applications to time and cost to start a business. Methodol. Comput. Appl. Probab. 2021;23:5–27. 10.1007/s11009-020-09801-1 Almarashi AM Jamal F Chesneau C : A new truncated muth generated family of distributions with applications. Complexity. 2021;2021(1):1211526. 10.1155/2021/1211526 Al AJN Resen IA Abdulwahab AM : The right truncated Xgamma-G family of distributions: Statistical properties and applications. AIP Conference Proceedings. AIP Publishing;2023. 10.1063/5.0161683 Abid S Abdulrazak R : [0, 1] Truncated Fréchet-Weibull and Fréchet Distributions. Int. J. Res. Ind. Eng. 2018;7(1):106–135. 10.22105/riej.2018.100865.1020 Khalaf AA : [0, 1] Truncated exponentiated exponential gompertz distribution: Properties and applications. AIP Conference Proceedings. AIP Publishing;2022. 10.1063/5.0121072 Altawil J : [0, 1] truncated lomax–lomax distribution with properties. J. Kufa Math. Comput. 2021;8(1):1–8. 10.31642/JoKMC/2018/080101 Cordeiro GM Ortega EMM Lemonte AJ : The exponential–Weibull lifetime distribution. J. Stat. Comput. Simul. 2014;84(12):2592–2606. Nasiru S : Serial Weibull Rayleigh distribution: theory and application. Int. J. Comput. Sci. Math. 2016;7(3):239–244. 10.1504/IJCSM.2016.077859 Mohammed MJ Hussein IH : Study of New Mixture Distribution [a]. International Conference Journal of Engineering and Apply Science ICEMASP in Turky. 2018. 10.36478/jeasci.2019.7566.7573 Mohammed AT Mohammed MJ Salman MD : The inverse exponential Rayleigh distribution and related concepts. Ital. J. PURE Appl. Math. Reference Source Shalan RN Alkanani IH : The Simulation Technique to Estimate the Parameters of Generalized Exponential Rayleigh Model. Iraqi. J. Sci. 2024;65(3):1502–1520. 10.24996/ijs.2024.65.3.27 Saleh HM Mohammed AT : Alpha power transformation family distributions: Properties and application to the exponential Weibull model. AIP Conference Proceedings. AIP Publishing;2024. 10.1063/5.0236939 Abid SH Abdulrazak RK : truncated Fréchet-G generator of distributions. Appl. Math. 2017;3:51–66. 10.5923/J.AM.20170703.03 Najarzadegan H Alamatsaz MH Hayati S : Truncated Weibull-G more flexible and more reliable than beta-G distribution. Int. J. Stat. Probab. 2017;6(5):1–17. 10.5539/ijsp.v6n5p1 Jamal F Bakouch H Nasir MA : A truncated general-G class of distributions with application to truncated Burr-G family. REVSTAT-Statistical J. 2021;19(4):513–530. Reference Source Gul A Mohsin M Adil M : A modified truncated distribution for modeling the heavy tail, engineering and environmental sciences data. PLoS One. 2021;16(4):e0249001. 33822800 10.1371/journal.pone.0249001 PMC8023488 Mohammed MJ Mohammed AT : Analysis of an Agricluture Data Using Markov Basis for Independent Model. J. Phys. Conf. Ser. IOP Publishing;2020;1530: p.012071. 10.1088/1742-6596/1530/1/012071, Mohammed AT Hussein IH : Density Estimation for Right Censored Data Using Hybrid Breslow and Semi-Symmetric Wavelet. J. Eng. Appl. Sci. 2019;14(1):242–246. 10.3923/jeasci.2019.242.246 Murthy DNP Xie M Jiang R : Weibull models. John Wiley & Sons;2004. 10.1002/047147326X.fmatter Mohammed MJ Mohammed AT Shalan RN : Application-of-Failure-time-of-50-components.[Data set]. Zenodo. 2025. 10.5281/zenodo.18071193 Mohammed MJ Mohammed AT Shalan RN ; Modified New [0, 1] - Right Truncated Exponential Rayleigh Distribution Structure and Application. Zenodo. 2026. 10.5281/zenodo.18119198 10.5256/f1000research.191964.r470691 Reviewer response for version 1 Husain Qasim Nasir 1 Referee https://orcid.org/0000-0002-9014-3261 Tikrit University, Baghdad, Iraq

Competing interests: No competing interests were disclosed.

 25 5 2026 Copyright: © 2026 Husain QN 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

My review of the research paper reveals the following: a serious proposal to expand and merge two basic distributions to obtain a new family, allowing the researcher to study the general and fundamental characteristics of the proposed family. There is also effective manipulation and modification of the parameters of the merged distributions to achieve greater flexibility. The results are supported by a simulation model, which is a strength of the work. The researcher needs to elaborate further in the conclusion to better illustrate the results and the subsequent analysis.

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?

Yes

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:

Mathematics, Mathematical Statistics, Data 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.

References : An Extension of Some Truncated Distributions Family With Application. Iraqi Journal of Science .2025; 10.24996/ijs.2025.66.1.25 317-328 10.24996/ijs.2025.66.1.25 10.5256/f1000research.191964.r485142 Reviewer response for version 1 Khaleel Mundher A 1 Referee Department of Mathematics, College of Computer Science and Mathematics, Tikrit University, Tikrit, Iraq

Competing interests: No competing interests were disclosed.

 21 5 2026 Copyright: © 2026 Khaleel MA 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

This paper presents an interesting and valuable contribution to the field of bounded lifetime distributions through the development of the modified [0,1]-Right Truncated Exponential Rayleigh Distribution. The manuscript demonstrates strong theoretical effort by deriving important statistical properties such as moments, entropy, quantile function, and order statistics, in addition to applying MLE and simulation analysis. The inclusion of practical applications and model comparison using information criteria strengthens the study’s applied significance. Moreover, the graphical analysis and flexibility of the proposed model indicate its potential usefulness in reliability and survival data analysis. Overall, the study has promising scientific value and can become a strong contribution after minor improvements in presentation and mathematical clarification.

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?

Yes

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?

Yes

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

Yes

Reviewer Expertise:

Mathematical Statistics

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.

 10.5256/f1000research.191964.r466937 Reviewer response for version 1 Sathar E.I.Abdul 1 Referee https://orcid.org/0000-0001-9032-8701 University of Kerala, Trivandrum, Kerala, India

Competing interests: No competing interests were disclosed.

 6 4 2026 Copyright: © 2026 Sathar EIA 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

Review Report

The manuscript proposes a new probability distribution called the [0,1] Right Truncated Exponential–Rayleigh Distribution ([0,1]-RTERD) derived from the exponential–Rayleigh distribution by applying right truncation on the interval ([0,1]). The paper derives the cumulative distribution function (CDF), probability density function (PDF), survival and hazard functions, moments, moment generating function, quantile function, skewness, kurtosis, and order statistics. The parameters are estimated using maximum likelihood estimation and the model is applied to real datasets for comparison with existing models.

The topic is relevant to statistical modeling of bounded lifetime data and proportion-type data. However, the manuscript requires significant improvements in presentation, mathematical clarity, and English language before it can be considered for Indexing.

Major Comments

1. The manuscript introduces a truncated version of the exponential–Rayleigh distribution, but the novel contribution is not sufficiently justified.

The authors should clearly explain:

Why the interval [0,1] truncation is necessary.

What advantages this model provides compared with other bounded distributions such as:

Beta distribution

Kumaraswamy distribution

Unit-Weibull distribution

Unit-Rayleigh distributions.

Currently the introduction only lists many generated families without explaining the research gap.

Recommendation

Add a paragraph explaining:

practical situations where [0,1] support is required

why existing models are insufficient

how the proposed distribution improves modeling flexibility.

2. The introduction cites several families of distributions but the discussion is very superficial.

Problems:

References are listed without explanation.

The relationship between these models and the proposed model is unclear.

Important literature on unit distributions and bounded distributions is missing.

 Recommendation

The authors should include literature on:

Unit distributions

Bounded lifetime models

Applications in reliability and proportion data.

3. Several derivations are unclear or incorrectly explained.

Examples:

The derivation of the moments (Equation 9) uses series expansion but the steps are not clearly justified.

Notation is inconsistent (for example (M_r^0)).

Some expressions contain typographical errors.

 For example:

The transformation (y=\delta x) is introduced without properly changing the integration limits.

The summation index appears incorrectly in the denominator in some equations.

 Recommendation

The mathematical derivations should be rewritten with:

clear intermediate steps

consistent notation

correct integral transformations.

4. Some formulas for skewness and kurtosis appear incorrect.

The paper states:

[

S.C = \frac{M_3^0}{(M_2^0)^{3/2}}

]

and

[

K.C = \frac{M_4^0}{(M_2^0)^2} - 3

]

However, skewness should normally use central moments, not raw moments.

This needs clarification.

5. Figures 1–4 show the CDF, PDF, survival, and hazard functions.

Problems:

The captions are vague (“Description of the figure…”).

Parameter values used in the plots are not clearly reported.

The figures do not demonstrate the flexibility of the distribution.

 Recommendation

Improve the figure captions and explain:

parameter combinations

shapes of hazard function

modeling implications.

6. The abstract claims that the model is applied to real datasets and compared with other distributions, but the details are not clearly presented.

Important missing elements:

dataset description

sample size

goodness-of-fit statistics

comparison tables (AIC, BIC, etc.)

graphical fitting analysis.

 Without these details, the practical value of the model cannot be assessed.

7. The manuscript contains many grammatical errors and awkward sentences.

Examples:

“Right – truncated [0,1] exponential Rayleigh distribution for lifetime is presented.”

“The peaks tend towards smaller values of (x) as (δ) increases.”

“A spatial case of eqn. (18)” (should be special case).

 These issues affect readability.

Recommendation

The manuscript requires major English editing by a fluent speaker or professional editing service.

Minor Comments

Symbols should be defined when first introduced.

Equation numbering is inconsistent.

Some references appear outdated.

Notation such as ((δ, λ)) should be consistently formatted.

Table captions should clearly describe the content.

The paper sometimes mixes lifetime terminology with bounded data terminology, which may confuse readers.

Strengths of the Paper

The study proposes a bounded lifetime distribution which may be useful for modeling proportions and reliability data.

Several statistical properties are derived.

The paper attempts both theoretical and applied analysis.

Parameter estimation using MLE is included.

Recommendation

Decision: Major Revision Required

The manuscript has potential but requires substantial revisions, including:

Clear explanation of novelty.

Improved literature review.

Correct and detailed mathematical derivations.

Stronger data analysis and model comparison.

Major language editing.

 After these improvements, the paper may be reconsidered for indexing.

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?

No

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:

Reliability Theory

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.