A New Probability Heavy-Tail Model for Stochastic Modeling under Engineering Data

El-Morshedy, M.; Eliwa, M. S.; Al-Bossly, Afrah; Yousof, Haitham M.

doi:https://doi.org/10.1155/2022/1910909

Journal of Mathematics

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Abstract Introduction Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1910909 | https://doi.org/10.1155/2022/1910909

A New Probability Heavy-Tail Model for Stochastic Modeling under Engineering Data

M. El-Morshedy,^1,2M. S. Eliwa,^3,4Afrah Al-Bossly,¹and Haitham M. Yousof⁵

Academic Editor: A. Ghareeb

Received04 Mar 2022

Accepted18 Apr 2022

Published22 Aug 2022

Abstract

The main aim of the paper is to propose and study a new heavy-tail model for stochastic modeling under engineering data. After studying and analyzing its mathematical properties, different classical estimation methods such as the ordinary least square, Cramér-von Mises, weighted least square, maximum likelihood, and Anderson–Darling estimation along with its corresponding left-tail and right-tail estimation methods are considered. Comprehensive numerical simulation studies are performed for comparing estimation methods in terms of some criterions. Three engineering and medical real-life data sets are considered for measuring the applicability flexibility of the new model and to compare the competitive models under uncensored scheme. Two engineering real-life data of them are also used to compare the classical methods. A modified Nikulin-Bagdonavicius goodness-of-fit is presented and applied accordingly for validation under censorship case. Finally, right censored lymphoma data set is analyzed under the modified statistic test for checking the validation of the reciprocal Weibull model in modeling the right censored data.

1. Introduction

The classic extreme value family contains the Gumbel (G) distribution (extreme value type I distribution); the reciprocal Weibull (RW) distribution (extreme value type II distribution), and the Weibull (W) distribution (extreme value type III distribution). The theory of extreme value (EV) theory focuses on the behavior of the block maxima or minima (see [1] for more details). The RW model is one of the most important distributions in modeling extreme values. The RW model was originally proposed by [1]. The EV family has many applications in ranging, accelerated life testing, floods, wind speeds, earthquakes, horse racing, rainfall, queues in supermarkets, and sea waves.

One can find more details about the RW model in the literature, for example, for applying the RW model in analyzing data of wind speed, see [2], for the beta-RW (B-RW) model, see [3], for the Marshall-Olkin RW (MORW) model, see [4], for the Weibull RW (W-RW), Weibull reciprocal Rayleigh (W-RR), and for the extended odd RW (EO-RW) family, among others. A random variable (RV) is said to have the RW distribution if its density function (PDF) and cumulative distribution function (CDF) are given byrespectively, where refers to the shape parameter. For , we obtain the reciprocal Rayleigh (RR) model. For , we have the reciprocal exponential (RE) model. Recently, [5] investigated and studied the GOLL-G family with regression models and applications to lifetime and red cell counts data. For any baseline CDF , the CDF of the GOLL-G class is given as

The PDF corresponding to (2) is given by

For , we get the OLL-G family (see[16]). For , we get the proportional reversed hazard rate G (PRHR-G) family (see [7]). The GOLLRW CDF is given by

The PDF corresponding to (4) is given by

The hazard rate function (HRF) for the new model can be got from .

In modeling EV real data sets, the GOLLRW model may be considered in the following applicable cases: modeling the “asymmetric heavy-tailed right skewed” data (see Section 5 and Section 6) and modeling the “asymmetric heavy-tailed right skewed” data in case of the EV data modeled for its first time. In reliability analysis and medical sciences, the OLLGRW model can be applied in modeling the stress data, fibers data, relief time data, and lymphoma data (see Section 5 and Section 6). For modeling breaking stress data, the Cramér-von Mises method performs well. For modeling the glass fibers data, the weighted least square method is the best one among all classical estimation methods. Using the validation approach of Bagdonavicius and Nikulin under right censored data, the modified chi-square goodness-of-fit (GOF) test helped us to say that the proposed GOLLRW model fits the lymphoma data. Table 1 (see the Appendix) provides some submodels of the GOLLRW model. As illustrated in Table 1, the new model generalizes eleven submodels, five of them are quite new. Some plots of the GOLLRW PDF and HRF are given in Figure 1 to illustrate some of its characteristics (see the Appendix). For simulation of this new model, we obtain the quantile function (QF) of (by inverting (4)), say , as

(a)

(b)

Equation (6) is used for simulating the new model. The novelty and gap of research are explained in more detail as follows:(i)The novel density in (5) can be “unimodal asymmetric and right skewed function” with various shapes.(ii)The HRF of the novel version can be upside-down (reversed U-HRF). This characteristic gives a bit advantage to the ROLLRW version for analyzing the uncensored data sets in which its HRF can be upside-down (reversed U-HRF).(iii)The proposed ROLLRW version is recommended for modeling the uncensored breaking stress data (which have many extreme values), the uncensored glass fiber data (which have some extreme values), and the uncensored relief time data (which have no extreme values).(iv)It is also recommended for modeling the data sets with its nonparametric. Kernel density estimation is bimodal and positive skewed with a very heavy tail and bimodal and positive skewed with light heavy tail and semisymmetric real-life data.(v)Moreover, the HRFs of the breaking stress data, glass fiber data, and relief time data are monotonically increasing; this property matches with the ROLLRW model which contains the increasing HRF.(vi)The skewness of the ROLLRW model (−26.23, 2.531), whereas the skewness of the standard baseline RW model (1.199, 5.565). Skewness of the ROLLRW model can be “negative” or “positive,” however, the skewness of the RW model can be only positive.(vii)The kurtosis of the ROLLRW model is located between and 10810.2, however the kurtosis of the RW model starts from 5.699 to 5436.5. Thus, the ROLLRW extension could be useful for leptokurtic, mesokurtic, and platykurtic data sets.(viii)The persuaders of estimating the unknown parameters of the ROLLRW version can be performed under Anderson–Darling, the maximum likelihood, ordinary least squares, weighted least squares, Cramér-von Mises, and left and right-tail Anderson–Darling methods.(ix)For modeling the uncensored breaking stress data, the Cramér-von Mises method performs well. Thus, it is recommended for future potential works in the field statistical modeling.(x)For modeling the uncensored glass fibers data, the weighted least square method is the best one among all classical estimation methods. Thus, it is recommended for future potential works in the field statistical modeling.(xi)Using the validation approach of Bagdonavicius and Nikulin statistic test under right censored real-life data, the modified chi-square goodness-of-fit statistic test helped us to recommend the proposed GOLLRW model for the right censored lymphoma data.

Thus, we are motivated to introduce and study the ROLLRW model for the abovementioned reasons.

2. Mathematical Properties

2.1. Useful Representations

Based on [5], the PDF in (5) can be expressed aswhereand is the PDF of the baseline RW model with scale parameter and shape parameter . Thus, the new density (6) can be reexpressed as a mixture of the RW PDFs. By integrating (7), the CDF of becomes where is the CDF of the RW distribution with scale parameter and shape parameter .

2.2. Moments and Incomplete Moments

The ordinary moment of is given by , then we obtainwhere and is the mean of . Numerical calculations and its corresponding analysis for the mean , variance , skewness , and kurtosis are calculated in Table 2 (for the GOLLRW) and Table 3 (for the RW) for some selected parameter values (see the Appendix). Based on Tables 2 and 3, we note that the , whereas the . Further, the spread for the is ranging from nearly 1.00 to nearly 10810.2, whereas the spread for the only varies from 5.699 to 5436.5 with the above parameter values. can be “negative” or “positive” however the can only be positive.

The incomplete moment, say , of the RV can be derived from (9) asthenwhere is the incomplete gamma function.

The first incomplete moment given by (11) with is

2.3. Moment Generating Function (MGF)

The MGF of can be derived from equation (7) as where is the MGF of the RW model with scale parameter and shape parameter , then

2.4. Residual Life (RLf) and Reversed Residual Life (RRLf) Functions and Their Moments

The moment of the RLf can be obtained by using or using . Therefore, the RLf can be written aswhere and Analogously, the moment of RRLf simply can be obtained from or from Then, the moment of the RRLf of becomeswhere

3. Classical Methods under Uncensored Schemes for Estimation

Consider the following classical estimation methods: maximum likelihood estimation (MLE), Cramér-von Mises estimation (CVME), ordinary least square estimation (OLSE), weighted least square estimation (WLSE), Anderson–Darling estimation (ADE), right-tail Anderson–Darling estimation (RTADE), and left-tail Anderson–Darling estimation (LTADE).

3.1. The MLE

Let be an observed random sample (ORS) from size from the GOLLRW distribution with parameters , and . Let be the vector of parameters. For obtaining the MLEs of , and , we derive the below log-likelihood function (LLF).

The score vector is available if needed and can be computed numerically.

3.2. The CVME

The CVME of the parameters , and are obtained via minimizing the function , wherewith respect to (WRT), the parameters , respectively, where and

The CVME of the parameters , and are derived by resolvingwhere , and .

3.3. The OLSE Method

Let denote the CDF of GOLLRW model and let be the ordered ORS. The OLSEs are derived by minimizingthen, we havewhere . The LSEs are derived by solvingwhere and are defined.

3.4. The WLSE Method

The WLSE can be gotten via minimizing WRT , and , wherewhere . The WLSEs are obtained by solvingwhere , and are defined.

3.5. The ADE Method

The ADE of , and are derived from minimizingi.e., the parameter estimates of , and follow by solving the following nonlinear system:

3.6. The RTADE Method

The RTADE of , and are obtained by minimizing

The parameter estimates of , and follow by solving the nonlinear equations.

3.7. The LTADE Method

The RTADE of , and are obtained by minimizing

The parameter estimates of , and follow by solving the nonlinear equations.

4. Uncensored Simulations for Comparing the Classical Methods

Simulation experiments are performed and then employed to assess and compare the classical methods. The assessment is based on N = data sets generated from the new model where , and , where

Figure 2 gives the density functions for three scenarios (see the Appendix). From Figure 2, it is seen that the PDF for all scenarios are asymmetric and right heavy tailed. The estimates of all methods are compared in terms of bias root mean − standard error mean of the absolute difference between the theoretical and the estimates (AAD-abs); and the maximum absolute difference between the true parameters and estimates (AAD-max). From Tables 4–6, we note that (see the Appendix):(i)The BIAS tends to as increases which shows that all estimators are “nonbiased.”(ii)The RMSE tends to as increases which shows the incidence of “consistency property.”

Generally, the MLE method is providing the better estimation with less RMSE compared to other classical methods for all sample sizes.(1)For “a₁ = 2.0, a₂ = 0.6, and a₃ = 0.8” (see Table 4), the MLE has lower RMSE where RMSE (a₁) is (0.24095, 0.16913, 0.09802, 0.07858). RMSE (a₂) is (0.05020, 0.03605, 0.02040, 0.01609). RMSE (a₃) is (0.08451, 0.05949, 0.03423, 0.02719).(2)For “a₁ = 0.5, a₂ = 1.5, and a₃ = 2.0” (see Table 5), the MLE has lower RMSE where RMSE (a₁) is (0.06269, 0.04207, 0.02363, 0.01875). RMSE (a₂) is (0.29559, 0.19878, 0.10739, 0.08458). RMSE (a₃) is (0.20936, 0.13836, 0.07680, 0.06002).(3)For “a = 0.5, b = 0.75, and θ = 1.5” (Table 6), the MLE has lower RMSE where RMSE (a₁) is (0.11128, 0.07842, 0.04342, 0.03293). RMSE (a₂) is (0.14310, 0.09885, 0.05514, 0.04348). RMSE (a₃) is (0.09804, 0.06704, 0.03675, 0.02747).

It is not easy to determine the worst classical estimation method since all other estimation methods perform well especially when n tends to

5. Real Data Modeling

In this section, we are interested with introducing three real data applications for comparing competitive models, two of them for comparing estimation methods.

5.1. Comparing Competitive Models under Uncensored Data

For illustrating the wide applicability of the new GOLLRW model, we consider the statistics: Cramér-Von Mises (); Anderson-Darling (); and Kolmogorov–Smirnov test (KS-test) and its corresponding value (P.V). The new model is compared with several common competitive models; Table 7 lists the competitive models and their corresponding abbreviations.

5.1.1. Breaking Stress Data

The 1^st data set is an uncensored data set consisting of 100 observations on breaking stress of carbon fibers (in Gba) given by [8]. Figure 3(a) gives the total time in test (TTT) plot (see [9]) for stress data set. It indicates that the empirical HRFs of data sets I is increasing. Figure 3(b) gives the box plot for discovering the outliers, Figure 3(c) gives the quantile-quantile (Q-Q) plot for checking the normality, and Figure 3(d) gives the nonparametric kernel density estimation (KrDE) exploring the density of the raw data. Figure 4 gives the P-P plot, estimated density (EPDF), ECDF, and EHRF for stress data set.

The statistics , KS-test, and P.V for all fitted models are presented in Table 8. The MLEs and corresponding standard errors (SEs) are given in Table 9. From Table 8, the GOLLRW model gives the smallest values, the , , and KS-test, and the biggest P.V statistics as compared to further RW models; therefore, the GOLLRW can be chosen as the best model. Figure 4 gives the estimated (E-PDF), estimated CDF (E-CDF), P-P plot, and estimated HRF (E-HRF) for stress data set.

5.1.2. Glass Fiber Data

The 2^nd data set is the generated data to simulate the strengths of glass fibers which was given by [10]. Figure 5 gives the TTT plot for fiber data set. It indicates that the empirical HRFs for fiber data sets are increasing. Figure 5(b) gives the box plot for discovering the outliers, Figure 5(c) gives the Q-Q plot for checking the normality, Figure 5(d) gives the nonparametric KeDE for exploring the density of the raw data. Figure 6 gives P-P plot, estimated density (EPDF), ECDF, and EHRF for glass fiber data. The , , KS-test, and P.V are listed in Table 10. The MLEs and SEs are given in Table 11. From Table 10, the GOLLRW model gives the lowest values, the , , and KS-test, and the biggest value of the P; therefore, the GOLLRW can be chosen as the best model. Figure 6 gives the E-PDF, E-CDF, P-P plot, and E-HRF for fiber data set.

5.1.3. Relief Time Data

The 3rd data set (called Wingo data, see [11]) is complete observed sample from a clinical trial describing relief times (in hours) for 50 arthritic patients. Figure 7 gives the TTT plot for relief time data set. It indicates that the empirical HRFs of relief time data set is increasing. Figure 7(b) gives the box plot for discovering the outliers, Figure 7(c) gives the Q-Q plot for checking the normality, and Figure 7(d) gives the nonparametric KeDE for exploring the density of the raw data. Figure 8 gives P-P plot, estimated density (EPDF), ECDF, and EHRF for relief times. The , , KS-test, and P.V are listed in Table 12. The MLEs and SEs are given in Table 13. From Table 12, the GOLLRW model gives the lowest values, the , , and KS-test, and the biggest value of the P.V; therefore, the GOLLRW can be chosen as the best model. Figure 8 gives the E-PDF, E-CDF, P-P plot, and E-HRF for relief time data set.

5.2. Comparing Estimation Methods under Uncensored Data

For comparing estimation methods via real data applications, we introduce two examples. The comparison is based on and . Tables 14–16 give the results for comparing estimation methods using breaking stress data and glass fiber data, respectively.

5.2.1. Example 1: The Breaking Stress Data

In this subsubsection and depending on breaking stress of carbon fibers (in Gba), the estimation methods will be compared. Table 14 lists the comparing results using breaking stress data. From Table 14, we note that the CVME method is the best method among all other methods with = 0.05864 and = 0.473, however all other methods performed well.

5.2.2. Example 2: The Glass Fiber Data

In this subsubsection and depending on the glass fiber data, the estimation methods will be compared. Table 15 lists the comparing results using breaking stress data. From Table 15, it is noted that the WLSE method is the best method among all other methods with = 0.04745 and = 0.39247, however all other methods performed well.

6. Censored Validation and Real Data Analysis

The statistic test is defined bywhere is the number of expected failures (NEF) in the grouped intervals and is the number of observed failures (NOF) in grouping intervals where

The elements of are defined bywhere refers to the cumulative HRF (CHRF) of the GOLLRW distribution. The quadratic form of the modified test statistic can be written aswhere matrices , , and are the estimated information matrices); for more details; see [12]. We have analyzed lymphoma data set consisting of times (in months) from diagnosis stage up to death for 31 individuals with the advanced non-Hodgkin’s lymphoma clinical symptoms, by using our model. Among these 31 observations, 11 of the times are censored, because those patients were still alive at the last time of follow-up: 2.5, 4.1, 4.6, 6.4, 6.7, 7.4, 7.6, 7.7, 7.8, 8.8, 13.3, 13.4, 18.3, 19.7, 21.9, 24.7, 27.5, 29.7, 30.1, 32.9, 33.5, 35.4, 37.7, 40.9, 42.6, 45.4, 48.5, 48.9, 60.4, 64.4, and 66.4 where denotes a censored observation. The statistic test is used to verify if the lymphoma data can be modeled by GOLLRW distribution. First, we have where the data are grouped into intervals. Table 16 gives the necessary calculus.

Second, we calculate the value of the :

For , the critical value which is larger than so we can say that the proposed GOLLRW model fits these data.

7. Concluding Remarks

In this paper, we introduced a new extension of the well-known reciprocal Weibull (RW) model, called the generalized odd log-logistic reciprocal Weibull (GOLLRW) model which is used for modeling the extreme values. The new model generalizes other eleven RW extensions, five of them are quite new. Some important mathematical properties of the new model are derived. The , whereas the . Further, the spread for the is ranging from nearly 1.00 to nearly 10810.2, whereas the spread for the only varies from 5.699 to 5436.5. can be “negative” or “positive,” however the can only be positive. We assessed the performance of seven estimation methods via some simulation experiments.

Three real data sets are presented for measuring the importance and flexibility of the new model and to compare the competitive models under uncensored scheme. The new model is better than some other important competitive models in modeling the breaking stress data, glass fiber data, and relief time data. The estimation methods are compared using two real data sets. For modeling the breaking stress of carbon fibers, the Cramér-von Mises method is the best method among all other methods, however all other methods performed well. For modeling the strengths of glass fiber data, the weighted least square method is the best method, however all other methods performed well. Finally, a modified Bagdonavicius-Nikulin GOF is presented and applied for validation under censorship case. Future works could be allocated for studying many new related extensions from other related aspects. The current study can be extended using neutrosophic statistics as future research (see [15–21]).

Data Availability

The 1st data set is an uncensored dataset consisting of 100 observations on breaking stress of carbon fibers (in Gba) given by [14]. The 2nd data set is generated to simulate the strengths of glass fibers which were given by [22]. The 3rd data set (called Wingo data) is complete observed sample from a clinical trial describing relief times (in hours) for 50 arthritic patients (see [23]). The 4th data set censored validation is given in Section 6.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this article.

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