Modified generalized Weibull distribution: theory and applications

This article presents and investigates a modified version of the Weibull distribution that incorporates four parameters and can effectively represent a hazard rate function with a shape resembling a bathtub. Its significance in the fields of lifetime and reliability stems from its ability to model both increasing and decreasing failure rates. The proposed distribution encompasses several well-known models such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh, and modified Weibull distributions. The paper derives key mathematical statistics of the proposed distribution, including the quantile function, moments, moment-generating function, and order statistics density. Various mathematical properties of the proposed model are established, and the unknown parameters of the distribution are estimated using different estimation techniques. Furthermore, the effectiveness of these estimators is assessed through numerical simulation studies. Finally, the paper applies the new model and compares it with various existing distributions by analyzing two real-life time data sets.

with H(x) denoted the cumulative hazard rate function satisfies the following conditions 1. lim x→0 H(x) = 0, 2. lim x→∞ H(x) = ∞, 3. H(x) is a differentiable non-negative and non-decreasing.By using Eq.(1), the generated cumulative density function (cdf) and probability density function (pdf) are, respectively, given by Some generalized distributions generated according to (2) and ( 3) are listed in Table 1.
Bagdonavicius and Nikulin 10 proposed an extension of the Weibull distribution, namely power generalized Weibull (PGW) distribution, and its cdf and pdf can be described as and and the relationship between cdf and pdf is given by respectively, where α and θ are two shape parameters and is a scale parameter.PGW distribution contains constant, monotone (increasing or decreasing), bathtub-shaped, and unimodal hazard shapes.For more details about this extension, see, for example, Bagdonavicius and Nikulin 11 , Voinov et al. 12 , and Kumar and Dey 13 .
In this research article, we introduce a novel statistical model called the modified power generalized Weibull (MPGW) distribution.Four parameters characterize the MPGW distribution and exhibit several significant properties.This distribution's probability density function (pdf) can assume different forms, including constant, monotonic (increasing or decreasing), and unimodal.Moreover, the hazard rate function (hrf) associated with the MPGW distribution can take on various shapes, such as constant, monotonic, bathtub, and upside-down bathtub.
We investigate several mathematical properties of the MPGW distribution and explore its applicability in different contexts.To estimate the model parameters, we employ various estimation techniques, including maximum likelihood estimation (MLE), the maximum product of spacing (MPS), least square estimators (LSE), and Cramer-von Mises estimators (CVE).These estimation methods enable us to determine the most suitable parameter values for the MPGW distribution based on the available data.
The proposed distribution was used in many fields of science such as engineering and bio-sciences as it can model many kinds of data because of the distribution's great flexibility.For more details about similar papers see 12,14 The rest of this paper is structured as follows.Section "The formulation of the MPGW distribution" described the new MPGW model and provided different distributional properties.Further, numerous statistical properties for the proposed distribution were introduced in Section "Statistical properties".In Section "Estimation methods", we established different estimation procedures for the unknown parameters of the suggested distribution.Monte Carlo simulation studies are performed in Section "Numerical simulation" to compare the proposed estimators.Finally, in Section "Real data analysis", two real data sets defined by the survival field are analyzed for validation purposes, and we conclude the article in Section "Conclusion".
Main contribution and novelty.This research paper presents a noteworthy advancement in the field of probability distributions by introducing a novel four-parameter generalization of the Weibull distribution.The (1) Table 1.Some generalized distributions of a mixture of the two chr functions.proposed generalization offers the ability to model a hazard rate function that exhibits a bathtub-shaped pattern.The bathtub-shaped hazard rate function is of great interest in various domains, as it accurately captures the characteristics of failure rates observed in certain real-world scenarios.To evaluate the efficacy of the newly proposed model, we conducted an empirical investigation using two distinct real-life time data sets.These data sets were carefully selected to encompass diverse applications and ensure the generalizability of the findings.

Name of the distribution
We could assess the model's effectiveness in practical applications by employing the proposed four-parameter generalized Weibull distribution and comparing its performance with several existing distributions.Through a comprehensive analysis of the results, valuable insights were obtained regarding the capabilities and advantages of the novel four-parameter generalized Weibull distribution when applied to real-world data sets.The comparison of the proposed model with existing distributions provided a rigorous evaluation framework, enabling a thorough understanding of its performance in different scenarios.This study contributes to the existing body of knowledge by demonstrating the applicability and usefulness of the new distribution in capturing the complexities of time-to-failure data.

The formulation of the MPGW distribution
The MPGW distribution is generated by using H 1 (x) of the PGW distribution and H 2 (x) of the exponential distribution in Eqs. ( 2) and (3).Its cdf and pdf can be defined as the following and the relationship between cdf and pdf can be written as where θ > 0 , , α, β ≥ 0 such that + β > 0 and α + β > 0.
The hazard rate function (hrf) of the MPGW model can be expressed as

Statistical properties
In this part of the study, we provided some mathematical properties of the MPGW distribution, especially moments, skewness, kurtosis, and asymmetry.
Behavior of the pdf of the MPGW distribution.The pdf limits of the MPGW distribution are From the pdf of the MPGW distribution, the first derivative of the pdf is .
Table 2. Some special models of the MPGW distribution.www.nature.com/scientificreports/where ψ(x) = (h(x)) 2 − h ′ (x) .It is clear that f ′ (x) and ψ(x) have the same sign, and ψ(x) has not an explicit solution.Therefore, we can discuss the following special cases which depend on θ and α: Case 1: For θ ≤ 1 and αθ ≤ 1 , ψ(x) is negative which means f (x) is decreasing in x Case 2: For θ = 1 , ψ(x) reduces to which has no solution for α ≤ 1 and the pdf becomes decreasing for all x.Case 3: For α = 1 , ψ(x) reduces to which has no solution for θ ≤ 1 and the pdf becomes decreasing for all x.Case 4: Forβ = 0 and θ = 1 , ψ(x) reduces to which has a solution for α>1 , therefore the mode (M) becomes Behavior of the hazard rate function of the MPGW distribution.The hrf limits of the MPGW distribution are www.nature.com/scientificreports/and The study of the shape of the hrf needs an analysis of the first derivative h ′ (x) and it can be described as where η(x) = θ − 1 + (αθ − 1)x θ .Clearly, h ′ (x) and η(x) have the same sign and η(x) has critical value at the point From η(x) , it can be noted that the hrf has different shapes written as: 1.If θ ≥ 1 , then h ′ (x) > 0 and h(x) are monotonically increasing.
2. If θ<1 , then the hrf is decreasing for x<x * and increasing forx>x * .Hence, the hrf has a bathtub shape.

Moments.
Theorem 1 For any r ∈ N , the rth raw moment of the MPGW model can be written as for or α=0, β>0 .  www.nature.com/scientificreports/Proof By the pdf (8) and the definition of the rth raw moment, we have In the general case, we suppose that , α and β>0 .Using the following expansion of e −βx given by then Eq. ( 12) is rewritten as x r+i e − 1+ x θ α dx and u = 1 + x θ α , we have By using the expansion of 1 − u −1/α (r+i+1)/θ −1 where u −1/α <1 , above integral is described as Hence, after some algebra, we get Hence, after some algebra, we obtain finally, substituting ( 14) and ( 15) into (13), we have which completes the proof.
According to the results given in theorem 3, the mean and the variance of the proposed model, respectively, are As well as the measures of skewness, kurtosis, and asymmetry of the MPGW are given, respectively, by and Table 3 shows some necessary MPGW measures for various parameter combinations computed using the R program.
From the values of Table 3 it can be deduced that ,

Estimation methods
Here, we considered four estimation techniques for constructing the estimation of the unknown parameters for MPGW model.The determination of the estimate parameters using different procedures has been made available to various authors such as [17][18][19] .
Maximum likelihood estimation and its asymptotics.Let {x 1 , . . ., x n } be a a random sample coming from MPGW(α, β, , θ) .Then, the corresponding log-likelihood function is described by with � = (α, β, , θ) .Consequently, with respect to α, β, , and θ and by taking the derivatives of ( 16), we can be determined the estimates αMLE , βMLE , ˆ MLE and θMLE and these estimates are given respectively by ( 16) and These estimates can be solved numerically using various approach methods, including Newton Raphson, bisection, or fixed point methods.
Least square estimation.Let x 1 , . . . ,x n be a random sample from MPGW(α, β, , θ) and represent the order statistics of the random sample from the MPGW model.The least-square estimator (LSE) which introduced by 20 ) of α, β, , θ , noted by αLSE , βLSE , ˆ LSE and θLSE ) can be described by minimizing Maximum product of spacings.For x 1 ≤ •

Numerical simulation
Here in this part of the work, we performed some results from simulation experiments so that you may assess how well the various estimating techniques provided in Section "Estimation methods" using different sample sizes, n = {100, 300, 500, 700, 1000} and different sets of initial parameters.After repeating the process K = 1000 , we generate different random samples from the suggested model.The following algorithm can be easily used to generate samples from the MPGW distribution 1.
Further, we compute the average values of biases (AB), mean square errors (MSEs), and mean relative errors (MREs) by the following equations where =(α, β, , θ ).All calculations were performed by using the R software version 4.1.2.Tables 4, 5 and 6 summarized the results of the simulation studies for the proposed model using the four estimation procedures.From the results, it can be concluded that as the sample size increases, all estimation methods of the proposed distribution approach to their initial guess of values.Furthermore, in all cases, the values of MSEs, and MREs tend to decrease.This ensures the consistency and asymptotically impartiality of all estimators.Additionally, by taking the MSE as an optimally criteria, we deduce that MLEs outperform alternative methods of estimate for the MPGWD.

Real data analysis
Through performing goodness-of-fit tests, we utilize two data sets to contrast the MPGW model with PGW distribution and the other four alternative existing models to see the effectiveness of the new model.The compared distributions: 1. Additive modified Weibull (AMW) distribution 4 with pdf defined as follows ( 19)  21 with pdf defined as follows  www.nature.com/scientificreports/ 3. Extended Weibull (EW) distribution 22 with pdf defined as follows 4. Flexible Weibull (FW) distribution 5 with pdf defined as follows 5. Kumaraswamy Weibull (KW) distribution 23 with pdf defined as follows 6. Beta Weibull (BW) distribution 24 with pdf defined as follows The first data set represents the recorded remission times given in months from bladder cancer patients, reported by Lee and Wang 25 .The ordered array of the data is  To assess the validity of the proposed model, we conducted several statistical tests and computed various criterion measures.Firstly, we computed the log-likelihood function (-L), then, we employed criterion measures such as the Akaike Information Criterion ( A 1 ) and the Bayesian Information Criterion ( B 1 ) to evaluate the performance of the model further.The model that yields the minimum values of these criteria is considered to be the most appropriate for the given data set.To complement the criterion measures, we also employed various test statistics, including the Cramér-von Mises (Cr), Anderson-Darling (An), and Kolmogorov-Smirnov (KS) tests.These tests assess the model's overall fit by comparing the observed data with the model's predicted values.The associated p-values obtained from these tests measure the statistical significance of the differences between the observed and predicted values.By considering these criterion measures and test statistics, we can comprehensively evaluate the validity of the proposed model.The model that exhibits the best fit, as indicated by the minimum values of the criterion measures and non-significant p-values from the test statistics, can be considered the most suitable for the given data set.
Tables 8 and 9, contain the values of criterion measure statistics for the fitted models by applying the two considered data sets.Based on these measures and along with the p-values of the proposed test statistics for each distribution, the MPGW model is the best candidate distribution for modeling the two data sets.The plots of the probability-probability (P-P) and quartile-quartile (Q-Q) of the suggested distributions using the two proposed data are shown in Figs. 3, 4, 5 and 6.This figure confirms this conclusion.
Figure 7 shows the curves of the pdfs for different fitting distributions using the first data set.Figure 8 shows the Curves of the pdfs for different fitting distributions using the second data set.Tables 10 and 11 contain The goodness of fit test for various fitting distributions by applying the first and second data sets, respectively.

Conclusion
This research paper introduces a novel distribution that involves compounding two cumulative hazard rate functions.We have derived a specific sub-model from the proposed distribution and established various mathematical properties related to it.We have applied four different estimation techniques to estimate the unknown parameters of our suggested model.Additionally, we have conducted simulation experiments to evaluate the effectiveness of these proposed estimation methods.Furthermore, we have analyzed two real engineering data sets to assess how https://doi.org/10.1038/s41598-023-38942-9

Case 5 : 6 :
For α = 1and β = 0 , ψ(x) reduces to which has a solution for θ>1 , therefore the mode becomes Case Forα = 1 , β = 0 and θ = 2 , ψ(x) reduces to in this case, the mode becomes For different parameter values, Fig.1depicts the pdf plots of MPGW distribution.The graphs show that the pdf of MPGW is decreasing and uni-modal which gives our proposed model the superiority for analyzing lifetime data.

Figure 1 .
Figure 1.Plot for PDF of the MPGW model for different values of the parameters.

Figure 2
Figure 2 displays the plot of hrf of MPGW model for multiple parameter values.The plots of hrf of MPGW are more efficient in modeling lifetime data.

Figure 2 .
Figure 2. Plot for PDF of the MPGW distribution for different values of the parameters.

Table 2
summarized several well-known lifetime distributions from the newly suggested distribution, which is quite flexible.

Table 3 .
Some statistical measures for MPGW using varied parameter values.
• • ≤ x n representing the ordered statistics random sample from MPGW distribution, the maximum product of the spacings estimation (MPS) estimators of the proposed model resulted by maximizing the following equation Cramer-von Mises minimum distance estimators.The Cramer-von Mises-type minimum distance estimators (CVEs) αCVE , βCVE , ˆ CVE and θCVE of α, β, , θ are described respectively by minimizing

Table 6 .
14e ABs, MSEs and associated MREs of the ( α, β, , θ)=(0.75, 0.6, 0.7, 0.4) considering different sample sizes.The second data set considered the values of the survival times given in days of guinea pigs infected with virulent tubercle bacilli, summarized by Bjerkedal14.The ordered array of the data is

Table 7
recorded different statistic measures for the two proposed data sets.

Table 7 .
Basic statistics for the two data.

Table 8 .
The MLEs and corresponding L, A 1 and B 1 values for different fitting models using first data.

Table 9 .
The MLEs and corresponding L, A 1 and B 1 values for different fitting models using second data.

Table 10 .
The goodness of fit test for various fitting distributions by applying the first data set.

Table 11 .
The goodness of fit test for various fitting distributions by applying the second data set.