The NCDF and NPDF functions, presented in Equations (7) and (8), respectively, are of great importance; however, this form makes integration, differentiation, and statistical analysis difficult. Therefore, these functions are expanded using power, exponential, and logarithmic series. This is because the original form of the functions contains very complex expressions that do not allow for a direct closed-form formula. Simplifying the mathematical operations and obtaining a flexible representation facilitates numerical analysis and application in estimation, calibration, and modeling. Therefore, the NCDF for NNOWW is expanded to the form: F NNOWW ( x N ) = 1 − ϖ e − α N q N x N β N (11) where ϖ = ∑ i , j , q = 0 ∞ ( − 1 ) i N + δ N + j N i N ! d i N δ N , j N λ N i N and d i N δ N , j N = j N − 1 ∑ w N = 1 j N [ w N ( i N δ N + 1 ) − j N ] w N + 1 for j N ≥ 0 and d i N k N , 0 = 1

The F NNOWW θ N ( x N ) has a form: F NNOWW θ N ( x N ) = ( 1 − e − λ N [ − ( 1 − e − α N x N β N ) . log ( e − α N x N β N ) ] δ N ) θ N (12)

In the same way, it can be expanded to form: F NNOWW θ N ( x N ) = Θ e − α N v N x N β N (13)

Where Θ = ∑ q N , l N , v N = 0 ∞ ( − 1 ) q N + l N + δ N + u N + v N l N ! ( θ N q N ) ( u N + 2 l N δ N v N ) d l N δ N , u N λ N l N q N l N , d l N δ N , u N = u N − 1 ∑ s N = 1 u N [ s N ( l N δ N + 1 ) − u N ] s N + 1 for u N ≥ 0

And d l N k N , 0 = 1

The NPDF for NNOWW can be expanded into the following series form: f NNOWW ( x N ) = ψ x N β N − 1 e − α N ( 1 + p N ) x N β N − φ x N β N − 1 e − α N ( 1 + t N ) x N β N (14)

Where ψ = ∑ i N , k N , n N , p N = 0 ∞ ( − 1 ) i N + 2 δ N + k N + n N + p N − 1 i N ! ( k N + 2 i N δ N + 2 δ N + n N − 1 p N ) d i N δ N + δ N − 1 , k N λ N i N + 1 δ N α N β N φ = ∑ i N , j N , t N = 0 ∞ ( − 1 ) i N + 2 δ N + j N + t N − 1 i N ! ( j N + 2 i N δ N + 2 δ N − 1 t N ) d i N δ N + δ N , j N λ N i N + 1 δ N α N β N such as d i N δ N + δ N − 1 , k N = k N − 1 ∑ s N = 1 k N [ s N ( i N δ N + δ N − 1 + 1 ) − k N ] s N + 1 for k N ≥ 0 and d i N δ N + δ N − 1 , 0 = 1 , and d i N δ N + δ N , j N = j N − 1 ∑ l N = 1 j N [ l N ( i N δ N + δ N + 1 ) − j N ] l N + 1 for j N ≥ 0 and d i N δ N + δ N , 0 = 1

The Neutrosophic quantile function (NQF) utilizes quantiles to determine the normal limits of values and detect extreme values within a distribution. It represents the mathematical inverse of the NCDF function F NNOWW ( x N ) , and is known as ³¹ : Q ( c ) = F NNOWW − 1 ( x N ) , c ∈ ( 0 , 1 )

That is, it gives the value x N at which the probability that the random variable X N is less than or equal to x N is c N . In other words: P ( X N ≤ Q ( c ) ) = c

Then the NQF for NNOWW has a form: Q ( c ) = [ log ( B B + W − 1 B e B ) α N ] 1 β N , B = ( − log ( 1 − p N ) λ N ) 1 δ N (15) where W − 1 is the lower Lambert W function.

The Table 1 shows the results of NQF for the NNOWW distribution across four sets of parameters (Set-1 to Set-4) shown below:

Set-1: λ N = [ 1.1,2.1 ] , δ N = [ 1.6,2.6 ] , α N = [ 1.8,2.8 ] , β N = [ 1.5,2.5 ]

Set-2: λ N = [ 1.2,2.2 ] , δ N = [ 1.5,2.5 ] , α N = [ 1.6,2.6 ] , β N = [ 1.7,2.7 ]

Set-3: λ N = [ 1.3,2.3 ] , δ N = [ 1.6,2.6 ] , α N = [ 1.4,2.4 ] , β N = [ 1.9,2.9 ]

Set-4: λ N = [ 1.4,2.4 ] , δ N = [ 1.6,2.6 ] , α N = [ 1.7,2.7 ] , β N = [ 1.6,2.7 ]

The results of Table 1 show how Q ( c ) values change at nine probability levels ranging from (0.1 to 0.9). Each result is given as an interval [lower, higher] representing the quantile’s neutrosophic values, where the lower limit reflects the degree of “truth” in the probability and the upper limit reflects the degree of “falseness” or vice versa, depending on the logical interpretation in a neutrosophic setting.

Comparing the values across the four sets, we note that set-3 produces the highest quantile values at each. c level, while set-1 produces the lowest. This suggests that the parameters in set-3 (remarkably the low α N and the high β N ) Result in a distribution that is more concentrated toward higher values; that is, the neutrosophic random variable tends to take larger values. In contrast, set-1 reflects a distribution more centered around smaller values.

The differences between the lower and higher values in each interval also reflect the amount of uncertainty in the probability estimation, which is the essence of the neutrosophic approach. The larger the gap between the two limits, the greater the degree of uncertainty. From this, we note that the differences are greater at the higher quantile values (0.8, 0.9), indicating that the extreme values carry the most significant amount of ambiguity.

Neutrosophic Moments function (NMF) are an extension of classical moments in statistics, taking into account degree of uncertainty. Their mathematical definition for a neutrosophic random variable X N , of order r , is ^{25, 32, 33} : μ N r = E ( x N r ) = ∫ − ∞ ∞ x N r f ( x N ) d x N (16)

Then the NMF for NNOWW has a form: μ N r = ψ ∫ 0 ∞ x N β N + r − 1 e − α N ( 1 + p N ) x N β N d x N − φ ∫ 0 ∞ x N β N + r − 1 e − α N ( 1 + t N ) x N β N d x N

By integrate above equation to get a final form: μ N r = Γ ( β N + r β N ) β N [ ψ { α N ( 1 + p N ) } β N + r β N − φ { α N ( 1 + t N ) } β N + r β N ] (17)

Then the first four NMF for NNOWW are: μ N 1 = Γ ( β N + 1 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 1 β N − φ { α N ( 1 + t N ) } β N + 1 β N ] (18) μ N 2 = Γ ( β N + 2 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 2 β N − φ { α N ( 1 + t N ) } β N + 2 β N ] (19) μ N 3 = Γ ( β N + 3 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 3 β N − φ { α N ( 1 + t N ) } β N + 3 β N ] (20) μ N 4 = Γ ( β N + 4 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 4 β N − φ { α N ( 1 + t N ) } β N + 4 β N ] (21)

From it, we can get the neutrosophic variance ( σ N 2 ),neutrosophic skewness ( γ 1 N ) and neutrosophic kurtosis ( γ 2 N ) of the NNOWW distribution as follows: σ N 2 = Γ ( β N + 2 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 2 β N − φ { α N ( 1 + t N ) } β N + 2 β N ] − ( Γ ( β N + 1 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 1 β N − φ { α N ( 1 + t N ) } β N + 1 β N ] ) 2 (22) γ 1 N = Γ ( β N + 3 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 3 β N − φ { α N ( 1 + t N ) } β N + 3 β N ] ( Γ ( β N + 2 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 2 β N − φ { α N ( 1 + t N ) } β N + 2 β N ] ) 3 2 (23) γ 2 N = Γ ( β N + 4 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 4 β N − φ { α N ( 1 + t N ) } β N + 4 β N ] ( Γ ( β N + 2 β N ) β N [ ψ { α N ( 1 + p N ) } β N + 2 β N − φ { α N ( 1 + t N ) } β N + 2 β N ] ) 2 − 3 (24)

Table 2 displays the first four NMF of the NNOWW distribution across different combinations of neutrosophic parameter values, neutrosophic variance, neutrosophic skewness, and neutrosophic kurtosis.

The Table 2 includes values for the intervals [lower, upper] that represent the moments of the distribution in an inexact setting, taking uncertainty into account. From the table, it is noted that increasing the parameters (especially β N and δ N ) leads to a change in the shape of the distribution, compressing it around the mean or spreading it out, depending on whether the values are small or large. Higher values of the moments reflect greater skewness in the data, while lower values indicate concentration around the median value.

Also, it is noted that the differences between the lower and upper bounds of each moment are wider when the probability values are high or when the parameters are distributed over larger intervals, indicating greater uncertainty in the estimate. This is consistent with the nature of the neutrosophic approach, where each probability value is expressed as a range rather than a single value, providing greater flexibility for modeling uncertain medical or environmental data.

Figure 4 presents a graphical representation of the NNOWW distribution’s NMF s across multiple parameter intervals. The figure illustrates how the neutrosophic values change at specific moments as the parameters are adjusted. The curves representing these moments define the significant impact of the parameters on the nature of the distribution.

The figure clearly shows that some intervals result in a more compressed distribution, meaning that the values are concentrated around a single point. In contrast, other intervals result in a more diffuse distribution, reflecting a greater variability in the values. This confirms that the NNOWW model has the flexibility to adapt to different data characteristics, especially data that is inherently noisy or fuzzy, such as the heartbeat data used in the application.

Table 3 presents the minimum and maximum NMF extracted from all analyzed intervals.

The Table 3 a quick numerical summary of the levels of uncertainty and skewness across NNOWW. It is clear that the differences between the extremes (lower and upper) gradually decrease at higher points, indicating that uncertainty is greater at the early points (the mean) and gradually decreases at higher points. The decrease in the value of the differences between the extremes indicates that the distribution is more stable in terms of its shape, skewness, and kurtosis, while retaining some flexibility in the location of the mean and variance.

The neutrosophic moment generating function (N.M.G.F) of any neutrosophic random variable X N is defined mathematically by the formula ^{34, 35} : M x N ( t N ) = E ( e t N x N ) = ∫ − ∞ ∞ e t N x N f ( x N ) d x N

Using exponential expansion for above equation, then we get a form: M x N ( t N ) = ∑ s N = 0 ∞ t N s N s N ! [ μ N r ] , r = 1 , 2 , 3 , 4

Using Equation (17) to get a form: M x N ( t N ) = ∑ s N = 0 ∞ t N s N s N ! [ Γ ( β N + r β N ) β N [ ψ { α N ( 1 + p N ) } β N + r β N − φ { α N ( 1 + t N ) } β N + r β N ] ] (25)

The neutrosophic Characteristic function (N.C.F) of any neutrosophic random variable X N is defined mathematically by the formula ³⁶ : Q x N ( t N ) = E ( e i t N x N ) = ∫ 0 ∞ e i t N x N f ( x N ) d x N

Using exponential expansion for above equation, then we get a form: Q x N ( t N ) = ∑ v N = 0 ∞ ( i t N ) v N v N ! [ μ N r ]

Using Equation (17) to get a form: Q x N ( t N ) = ∑ v N = 0 ∞ ( i t N ) v N v N ! [ Γ ( β N + r β N ) β N [ ψ { α N ( 1 + p N ) } β N + r β N − φ { α N ( 1 + t N ) } β N + r β N ] ] (26)

Neutrosophic Incomplete Moments are an extension of classical Incomplete moments in statistics, taking into account degree of uncertainty. Their mathematical definition for a neutrosophic random variable X N , of order r , is ^{25, 32, 33} : M r ( y ) = ∫ − ∞ y x N r f ( x N ) d x N (27)

Then the Neutrosophic Incomplete Moments for NNOWW from above equation and Equation (14) has a form: M r ( y ) = ψ ∫ 0 y x N β N + r − 1 e − α N ( 1 + p N ) x N β N d x N − φ ∫ 0 y x N β N + r − 1 e − α N ( 1 + t N ) x N β N d x N

By integrate above equation to get a final form: M r ( y ) = 1 β N [ ψΓ ( β N + r β N , α N ( 1 + p N ) y β N ) { α N ( 1 + p N ) } β N + r β N − φΓ ( β N + r β N , α N ( 1 + t N ) y β N ) { α N ( 1 + t N ) } β N + r β N ] (28)

The Neutrosophic Rényi entropy is a generalization of the Shannon entropy. It provides a parametric family of entropy measures that can be adjusted using a parameter θ N , which controls the sensitivity to events with small probabilities. Let f ( x N ) be a NPDF for any distribution then the Neutrosophic Rényi entropy is defined as ³⁷ : R θ N ( f ) = 1 1 − θ N log ( ∫ 0 ∞ f θ N ( x N ) d x N ) , θ N > 0 & θ N ≠ 1 (29)

From Equations (29) and (14) we get: R θ N ( f ) = 1 1 − θ N log ( ∫ 0 ∞ ( ψ x N β N − 1 e − α N ( 1 + p N ) x N β N − φ x N β N − 1 e − α N ( 1 + t N ) x N β N ) θ N d x N )

Using the Binomial Series ( ψ x N β N − 1 e − α N ( 1 + p N ) x N β N − φ x N β N − 1 e − α N ( 1 + t N ) x N β N ) θ N = ∑ k = 0 ∞ ( − 1 ) k ( θ N k ) ψ θ N − k x N β N + θ N − k − 1 e − α N ( θ N − k ) ( 1 + p N ) x N β N φ k x N β N + k − 1 e − α N k ( 1 + t N ) x N β N ( ψ x N β N − 1 e − α N ( 1 + p N ) x N β N − φ x N β N − 1 e − α N ( 1 + t N ) x N β N ) θ N = ∑ k = 0 ∞ ( − 1 ) k ( θ N k ) ψ θ N − k x N 2 β N + θ N − 2 e − α N [ θ N ( 1 + p N ) + k ( 1 + t N ) ] x N β N φ k

Then we get: R θ N ( f ) = 1 1 − θ N log ( ∑ k = 0 ∞ ( − 1 ) k ( θ N k ) ψ θ N − k φ k ∫ 0 ∞ x N 2 β N + θ N − 2 e − W x N β N d x N )

Where W = α N [ θ N ( 1 + p N ) + k ( 1 + t N ) ] the final form is: R θ N ( f ) = 1 1 − θ N log ( ∑ k = 0 ∞ ( − 1 ) k ( θ N k ) ψ θ N − k φ k 1 β N W − 2 β N + θ N − 1 β N Γ ( β N + θ N − 1 β N ) ) (30)

Maximum Likelihood Estimation (MLE) is a fundamental inferential technique employed to estimate the unknown parameters of a statistical model. It operates by identifying the parameter values that maximize the likelihood function, which quantifies the probability of observing the given dataset under the assumed model structure.

Formally, consider a random sample X N = ( x N 1 , x N 2 , … , x Nn ) drawn from a population characterized by a probability density function f ( x N | γ N ) , where γ N Denotes the vector of unknown parameters ( γ N = ( λ N , δ N , α N , β N ) ). The likelihood function L ( γ N | x N ) is then defined as ³⁸ : L ( γ N | x N ) = ∏ i = 1 n f ( x Ni | γ N )

From Equation 8 the MLE for NNOWW has a form: L ( γ N | x N ) = ∏ i = 1 n δ N α N β N x Ni β N − 1 e − α N x Ni β N [ 1 − e − α N x Ni β N e − α N x Ni β N − log ( e − α N x Ni β N ) ] [ − ( 1 − e − α N x Ni β N ) log ( e − α N x Ni β N ) ] 1 − δ N × e − λ N [ − ( 1 − e − α N x Ni β N ) . log ( e − α N x Ni β N ) ] δ N

The objective of MLE is to find the parameter vector γ ̂ N that maximizes this function, that is: γ ̂ N = arg max γ N L ( γ N | x N )

In practice, for computational convenience, the natural logarithm of the likelihood function, known as the log-likelihood, is often used: l ( γ N | x N ) = nlog ( λ N ) + nlog ( δ N ) + nlog ( α N ) + nlog ( β N ) − ∑ i = 1 n α N x iN β N + ( β N − 1 ) ∑ i = 1 n log ( x Ni ) + ∑ i = 1 n log [ 1 − e − α N x iN β N e − α N x iN β N − log ( e − α N x iN β N ) ] + ( δ N − 1 ) ∑ i = 1 n log [ − ( 1 − e − α N x iN β N ) . log ( e − α N x iN β N ) ] − λ N ∑ i = 1 n [ − ( 1 − e − α N x iN β N ) . log ( e − α N x Ni β N ) ] (31)

The estimation process involves solving the system of equations obtained by setting the partial derivatives of the log-likelihood function in Equation 31 concerning each parameter to zero. This yields the maximum likelihood estimates of γ N .

Least Squares Estimation (LSE) is used to estimate the parameters of a linear regression model by minimizing the sum of the squared differences between the observed values and the values predicted by the model, and defined as: φ ( γ N ) = ∑ i = 1 n [ F ( x Ni ) − i n + 1 ] 2

Then the LSE for NNOWW form above equation and equation has a form: φ ( γ N ) = ∑ i = 1 n [ 1 − e − λ N [ − ( 1 − e − α N x Ni β N ) . log ( e − α N x Ni β N ) ] δ N − i n + 1 ] 2 (32)

Weighted Least Squares (WLS) is a type of extension of LSE used when the assumption of constant variance (homoscedasticity) in the error terms is violated. It assigns weights to each observation to address heteroscedasticity. and defined as: W ( γ N ) = ∑ i = 1 n W i [ F ( x Ni ) − i n + 1 ] 2 where W i is the weight to each observation defined by form: W i = ( n + 1 ) 2 ( n + 2 ) i ( n − i + 1 )

Then the LSE for NNOWW form above equation and equation has a form: W ( γ N ) = ∑ i = 1 n W i [ 1 − e − λ N [ − ( 1 − e − α N x Ni β N ) . log ( e − α N x Ni β N ) ] δ N − i n + 1 ] 2 (33)

Equations 32 and 33 are solved in the same manner as Equation 31: by differentiating the four NNOWW parameters and then setting the derivatives to zero.

Simulation is a fundamental step in evaluating the performance of statistical estimation methods under controlled experimental conditions. It allows for testing the accuracy, reliability, and efficiency of the various parameter estimators presented in Section 4 when they are repeatedly drawn from a known distribution with predetermined parameters.

In this study, a Monte Carlo simulation was conducted to evaluate the behavior of the proposed NNOWW using various sample sizes. The goal was to measure the ability of each estimation method to retrieve the actual values of the parameters when applied to randomly generated data, while simultaneously reflecting the nature of neutrosophic uncertainty.

Several performance indicators were used, including the arithmetic mean of the estimates to measure the average estimated values across iterations; bias ³⁹ to measure the difference between the estimated and actual values; mean square error (MSE) ⁴⁰ to combine the variance and bias of the estimate; and root mean square error (RMSE) ^{41, 42} to directly measure the average error in the estimate. The simulation was conducted for four different sample sizes (30, 60, 120, and 240 to 1000 iteration) to analyze and improve the effectiveness of the estimation methods as the data volume increases. Neutrosophic intervals were used for the parameters instead of exact values to represent the uncertain nature of the data, in line with the neutrosophic logic adopted in the study.

The results shown in Table 4 reflect the efficiency default parameter values are given to illustrate the estimation efficiency of each method and highlight the strengths and shortcomings of each, enhancing the reliability of the proposed distribution and providing a reliable scientific basis for future applications of ambiguous data, as shown below.

From Table 4, it is clear that. •

Size 30: At this small sample size, the estimates show wide variation among the three methods, especially in the MSE and RMSE metrics. The maximum likelihood method performs averagely, but the LSE and WLSE show greater variation in estimates, reflected in the high MSE values. LSE produces the highest bias, indicating that the method is unstable in small samples, and WLSE suffers from dispersion in estimates, especially in the second and third estimates. The significant difference between the lower and upper bounds for all methods reflects the high degree of uncertainty in the estimate at a small sample size.

Thus, as the sample size increases, all methods improve; however, the MLE method is the most balanced in terms of reducing bias and variance across all stages.

Based on the simulation results in the table, the graphs in Figure 5 confirm the numerical results in the simulation table, providing a more precise visual representation.

The results show that when the sample size is small, the differences between methods are significant, and the estimates are unreliable. As the sample size increases, the differences narrow, and the estimates stabilize, especially for MLE.

For λ N , in small samples ( n = 30 ), the lines appear scattered and wide, indicating high volatility and instability in the estimate, especially for LSE and WLSE. As the sample size increases, the lines begin to compress towards specific values, indicating improved accuracy and reduced dispersion. For the δ N , the graphs show that WLSE significantly improved this estimate at sizes of 120 and 240 , and began to outperform LSE in reducing volatility. For α N , this particular plot illustrates the parameters most significantly affected by the increase in the sample size at n = 30 . The three lines are very far apart. At n = 120 and n = 240 , we find that the MLE approaches a clear horizontal line, indicating a quasi-stationary and highly accurate estimate. The WLSE gradually approaches but remains slightly fluctuating at the upper limit.

The applied part of the study focused on testing the effectiveness of the NNOWD by applying it to real-world data. The pulse rate is generally regarded as a discrete variable. The proposed test is applied using the pulse rate counts of 11 patients ⁴³ was used as a real-world example to verify the suitability of the proposed distribution in describing data characterized by uncertainty. Before verifying the data, it is checked that the data meets the properties of neutral logic. Table 5 shows the actual values of the data (Intervals), the three neutral components (T, F, I), and the condition “ T + F + I = 1 ” to verify the validity of the neutral formula.

The following figures illustrate the data used, the nature of its analysis, and the components of its analysis.

All values satisfy the sum = 1 condition, indicating the accuracy of the components’ calculations. Some negative values appear in F and I, indicating that the values fall outside the range [ 0 , 1 ] if no restriction or normalization is applied. However, the study analyzes them at the level of actual values, not relative values. The table reflects the methodology’s ability to extract the components of knowledge, ignorance, and uncertainty from real data.

All figures are used to validate the three components of neutrosophic logic (T, F, I) using graphs and to illustrate the behavior of these values across the data.

Figure 6 plots the highest and lowest values for each observation, illustrating the range of data used in the T, F, I analysis.

Figure 7 is a graphical representation of the distribution of T, F, and I across all observations, showing that the values are concentrated within a specific range for each component and illustrating the variance between them.

Figure 8 plots the sum of T, F, and I for each observation. This plot is a crucial step in validating the neutrosophic analysis, as it demonstrates that all values align with the value 1, a prerequisite for neutrosophic analysis. This validation ensures the accuracy and reliability of the methodology.

Figure 9 plots Trend plot of Truth value for Heart Pulse Count Data.

After verifying the conditions of neutrosophic logic on the data used, the effectiveness of the proposed distribution is confirmed by comparing it with five other distributions because it’s have the same number of parameters, namely Neutrosophic Exponential Generalized Exponential Weibull (NEGW), Neutrosophic Kumaraswamy Weibull (NKuW), Neutrosophic Log-Gamma Weibull (NLGamW), Neutrosophic Beta Weibull (NBEW), and Neutrosophic Gompertz Weibull (NGoW). The comparison proceeds through three stages: first, using information criteria such as Akaike (AIC), ⁴⁴ Corrected Akaike (CAIC), ⁴⁵ Hann-Quinn (HQIC), ⁴⁶ and Bayes criterion (BIC) ⁴⁷ ; second, employing four statistical measures: Kolmogorov-Smirnov (KS), Anderson-Darling (A), Cramér-von Mises (W), and p-value; and finally, verifying the parameters estimated by the maximum likelihood method for each distribution. the analysis results which was shown in Table 6, Table 7 and Table 8.