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

Authors: M. El-Morshedy; M. S. Eliwa; Afrah Al-Bossly; Haitham M. Yousof
Journal: Journal of Mathematics
 (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.


---

## Body

## 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) Y is said to have the RW distribution if its density function (PDF) and cumulative distribution function (CDF) are given by(1)ga3y=a3y−a3+1exp−y−a3|y≥0,Ga3y=exp−y−a3|y≥0,respectively, where a3>0 refers to the shape parameter. For a3=2, we obtain the reciprocal Rayleigh (RR) model. For a3=1, 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 GΦ¯y, the CDF of the GOLL-G class is given as(2)Fa1,a2,Φ¯y=GΦ¯ya1a2GΦ¯ya1a2+1−GΦ¯ya2a1|a1>0,a2>0,Φ¯∈R.The PDF corresponding to (2) is given by(3)fa1,a2,Φ¯y=a1a2gΦ¯yGΦ¯ya1a2−11−GΦ¯ya2a1−1GΦ¯ya1a2+1−GΦ¯ya2a12|a1>0,a2>0,Φ¯∈R.Fora2=1, we get the OLL-G family (see[16]). For a1=1, we get the proportional reversed hazard rate G (PRHR-G) family (see [7]). The GOLLRW CDF is given by(4)FV¯y=exp−a1a2y−a3exp−a1a2y−a3+1−exp−a2y−a3a1|V¯=a1,a2,a3.The PDF corresponding to (4) is given by(5)fV¯y=a1a2a3y−a3+1exp−a1a2y−a31−exp−a2y−a3a1−1exp−a1a2y−a3+1−exp−a2y−a3a12.The hazard rate function (HRF) for the new model can be got fromfV¯y/FV¯y.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 Section5 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 Y (by inverting (4)), say yu=Qu=F−1u, as(6)yu=−lnu/1−u1/a11+u/1−u1/a11/a2−1/a3.Table 1
Especial members of the GOLLRW model.
Na1a2a3Reduced modelReduced CDFAuthor1a1a22GOLL-RRexp−a1a2y−2/exp−a1a2y−2+1−exp−a2y−2a1New2a1a21GOLL-RExexp−a1a21/y/exp−a1a2y−1+1−exp−a2y−1a1New31a2a3PRHR-RWexp−a2y−a3[13]41a22PRH-RRexp−a2y−2[13]51a21PRHR-RExexp−a2y−1[13]6a11a3OLLRWexp−a1y−a3/exp−a1y−a3+1−exp−y−a3a1[11]7a112OLL-RRexp−a1y−2/exp−a1y−2+1−exp−y−2a1[11]8a111OLL-RExexp−a1y−1/1−exp−y−1a1+exp−a1y−1[11]911a3RWexp−y−a3[1]10112RRexp−y−2[1]11a1a2a3RExexp−y−1[14]Figure 1
Plots of the GOLLRW PDF (a) and GOLLRW HRF (b).
(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 between1 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 as(7)fV¯y=∑κ4=0∞ϒκ4π1+κ4y;a3,where(8)ϒκ4=a1a21+κ4∑κ1,κ2,κ3=0∞∑κ4=0κ3−1κ2+κ4+κ3−2κ1−a1κ1+1κ2a1a2κ1+1+a2κ2−1κ3κ3κ4,and π1+κ4y;a3 is the PDF of the baseline RW model with scale parameter 1+κ4a3 and shape parameter a3. Thus, the new density (6) can be reexpressed as a mixture of the RW PDFs. By integrating (7), the CDF of Y becomes FV¯y=∑κ4=0∞ϒκ4∏1+κ4y;a3, where ∏1+κ4y;a3 is the CDF of the RW distribution with scale parameter κ4a3 and shape parameter a3.


### 2.2. Moments and Incomplete Moments

Thepth ordinary moment of Y is given by μp′=EYP=∫−∞∞ypfV¯ydy, then we obtain(9)μp′|a3>p=Γ1−pa3∑κ4=0∞ϒκ41+κ4p/a3,where Γ1+∇|∇∈R+=∇!=∏h=0∇−1∇−h and Ey=μ1′|a3>1=Γ1−1/a3∑κ4=0∞ϒκ41+κ41/a3 is the mean of Y. Numerical calculations and its corresponding analysis for the mean EX, variance VarX, skewness SkewY, and kurtosis KurY 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 SkewGOLLRWY∈−26.23,2.531, whereas the SkewRWY∈1.199,5.565. Further, the spread for the KurGOLLRWY is ranging from nearly 1.00 to nearly 10810.2, whereas the spread for the KurRWY only varies from 5.699 to 5436.5 with the above parameter values. SkewGOLLRWY can be “negative” or “positive” however the SkewRWY can only be positive.Table 2
E (X), Var (X), Skew (Y), and Kur (Y) of the GOLLRW model.
a1a2a3E (Y)Var (Y)Skew (Y)Kur (Y)5351.3451010.005056890.63180445.0616721530101.4577960.000161840.16973004.2546021010101.3063760.000292750.25347834.3215133331.6628680.064969401.4463169.4547366241.3076240.005178790.5848004.9700884101.258.8805796.720131002.53074331.179772011.251.3432700.004982800.35178774.51182655251.0827530.000128640.41779944.4685875422.4357830.108744301.0983947.3500310422.4105620.025297350.51680144.86249953.251101.0142445.922422 × 10−6−7.363551∗3226.47353.251101.0149285.850545 × 10−6−1.786044∗911.30185322.109450.081558231.09839407.3500331.6415840.021326040.82600435.83384341.4492120.009234300.70256235.31762851.3451010.005056890.63180445.061672101.1593850.000927860.49519554.687103501.0299632.945824 × 10−5−2.294597∗512.6364551.0272012.428456 × 10−5−3.185613∗747.5396601.0249052.037847 × 10−5−4.239300∗1052.191651.0229671.735751 × 10−5−5.470000∗1438.862751.0198741.305848 × 10−6−8.510591∗2509.386801.018621.149769 × 10−6−10.34218∗3219.911001.0148677.449114 × 10−6−19.95313∗7553.1951101.0135066.209712 × 10−6−26.22742∗10810.18250.52.50.8778620.000338230.20405703.03838411.1583450.000589180.18031594.16134152.2050880.002135230.17789894.276042102.9096310.003717660.17789954.276013505.5389310.013472410.17789794.2760891007.3086640.023456830.17789964.276008521.52.0727920.144995601.41954909.958146102.0380830.032417620.63316235.224646152.0317790.014117390.41391294.630934202.0295820.007884810.30834094.43794302.0280150.003486640.20457754.304321502.0272140.001251940.12244614.237321602.0270760.000869010.10199764.225757705.10 × 10−54.109339001.00015801.000521Table 3
E (X), Var (X), Skew (Y), and Kur (Y) of the RW model.
a3E (Y)Var (Y)Skew (Y)Kur (Y)4.011.2245880.26851075.5650775436.5964.11.2173870.24911105.236318529.59174.251.2063320.22121624.790364204.83094.51.1901510.18425634.23883798.8014551.164230.13376143.53507248.0915161.1287870.07995782.80556624.6781271.1057670.05327192.42509717.5340281.0896520.03807442.1892714.16589101.0686290.02226241.91033910.97857121.0555470.01460851.7498099.461719151.0431670.00885811.6052458.282494201.0314530.004732761.4738847.333494251.0247320.002940271.4004866.852401301.0203740.002002721.3535666.562309351.0173210.001451511.3209686.368528401.0150630.001100171.2969986.229997451.0133260.000862541.2786286.126068501.0119470.000694361.2640996.045237601.0098990.000477791.2425795.927632751.0078740.000303011.2213745.814377801.0073720.000265721.2161225.786696901.0065370.000209161.2074105.7410881001.0058720.000168921.2004785.7051761011.0058130.000165541.1998625.7019951021.0057550.000162271.1992585.69888Thepth incomplete moment, say Ip,Yt, of the RV Y can be derived from (9) as(10)Ip,Yt=∫−∞typfV¯ydy=∑κ4=0∞ϒκ4∫−∞typπ1+κ4y;a3dy,then(11)Ip,Yt|a3>p=∑κ4=0∞ϒκ41+κ4p/a3γ1−pa3,1+κ41ta3,where γ∇,p is the incomplete gamma function.(12)γ∇,p|∇≠0,−1,−2,…=∫0pt∇−1exp−tdt=∑κ4=0∞p∇+κ4κ4!∇+κ4.The first incomplete moment given by (11) with p=1 is(13)I1,Yt|a3>1=∑κ4=0∞wκ41+κ4a3γ1−1a3,1+κ41ta3.


### 2.3. Moment Generating Function (MGF)

The MGF ofY can be derived from equation (7) as MYt=∑κ4=0∞ϒκ4MY,1+κ4t;a3, where MY,1+κ4t;a3 is the MGF of the RW model with scale parameter 1+κ4a3 and shape parameter a3, then(14)MYt|a3>p=Γ1−pa3∑κ4=0∞∑p=0∞tpp!ϒκ41+κ4p/a3.


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

Themth moment of the RLf can be obtained by using lm,Yt=EY−tm|Y>t,m∈ℕ or using lm,Yt=1/1−FV¯t∫t∞Y−tmdFV¯y. Therefore, the RLf can be written as(15)lm,Yt=Γ1−m/a3,1+κ41/ta31−FV¯t∑κ4=0∞cκ411+κ4m/a3,∀a3>m,where cκ41=ϒκ4∑p=0mmp−tm−p,Γ∇,p|y>0=∫p∞t∇−1exp−tdt, and Γ∇,p+γ∇,p=Γ∇. Analogously, the mth moment of RRLf simply can be obtained from Lm,Yt=Et−Ym|Y≤t,t>0andm∈ℕ or from Lm,Yt=1/FV¯t∫0tt−YmdFV¯y. Then, the mth moment of the RRLf of Y becomes(16)Lm,Yt=γ1−m/a3,1+κ41/ta3FV¯t∑κ4=0∞cκ421+κ4m/a3,∀a3>m,where cκ42=ϒκ4∑p=0m−1pmptm−p.


## 2.1. Useful Representations

Based on [5], the PDF in (5) can be expressed as(7)fV¯y=∑κ4=0∞ϒκ4π1+κ4y;a3,where(8)ϒκ4=a1a21+κ4∑κ1,κ2,κ3=0∞∑κ4=0κ3−1κ2+κ4+κ3−2κ1−a1κ1+1κ2a1a2κ1+1+a2κ2−1κ3κ3κ4,and π1+κ4y;a3 is the PDF of the baseline RW model with scale parameter 1+κ4a3 and shape parameter a3. Thus, the new density (6) can be reexpressed as a mixture of the RW PDFs. By integrating (7), the CDF of Y becomes FV¯y=∑κ4=0∞ϒκ4∏1+κ4y;a3, where ∏1+κ4y;a3 is the CDF of the RW distribution with scale parameter κ4a3 and shape parameter a3.


## 2.2. Moments and Incomplete Moments

Thepth ordinary moment of Y is given by μp′=EYP=∫−∞∞ypfV¯ydy, then we obtain(9)μp′|a3>p=Γ1−pa3∑κ4=0∞ϒκ41+κ4p/a3,where Γ1+∇|∇∈R+=∇!=∏h=0∇−1∇−h and Ey=μ1′|a3>1=Γ1−1/a3∑κ4=0∞ϒκ41+κ41/a3 is the mean of Y. Numerical calculations and its corresponding analysis for the mean EX, variance VarX, skewness SkewY, and kurtosis KurY 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 SkewGOLLRWY∈−26.23,2.531, whereas the SkewRWY∈1.199,5.565. Further, the spread for the KurGOLLRWY is ranging from nearly 1.00 to nearly 10810.2, whereas the spread for the KurRWY only varies from 5.699 to 5436.5 with the above parameter values. SkewGOLLRWY can be “negative” or “positive” however the SkewRWY can only be positive.Table 2
E (X), Var (X), Skew (Y), and Kur (Y) of the GOLLRW model.
a1a2a3E (Y)Var (Y)Skew (Y)Kur (Y)5351.3451010.005056890.63180445.0616721530101.4577960.000161840.16973004.2546021010101.3063760.000292750.25347834.3215133331.6628680.064969401.4463169.4547366241.3076240.005178790.5848004.9700884101.258.8805796.720131002.53074331.179772011.251.3432700.004982800.35178774.51182655251.0827530.000128640.41779944.4685875422.4357830.108744301.0983947.3500310422.4105620.025297350.51680144.86249953.251101.0142445.922422 × 10−6−7.363551∗3226.47353.251101.0149285.850545 × 10−6−1.786044∗911.30185322.109450.081558231.09839407.3500331.6415840.021326040.82600435.83384341.4492120.009234300.70256235.31762851.3451010.005056890.63180445.061672101.1593850.000927860.49519554.687103501.0299632.945824 × 10−5−2.294597∗512.6364551.0272012.428456 × 10−5−3.185613∗747.5396601.0249052.037847 × 10−5−4.239300∗1052.191651.0229671.735751 × 10−5−5.470000∗1438.862751.0198741.305848 × 10−6−8.510591∗2509.386801.018621.149769 × 10−6−10.34218∗3219.911001.0148677.449114 × 10−6−19.95313∗7553.1951101.0135066.209712 × 10−6−26.22742∗10810.18250.52.50.8778620.000338230.20405703.03838411.1583450.000589180.18031594.16134152.2050880.002135230.17789894.276042102.9096310.003717660.17789954.276013505.5389310.013472410.17789794.2760891007.3086640.023456830.17789964.276008521.52.0727920.144995601.41954909.958146102.0380830.032417620.63316235.224646152.0317790.014117390.41391294.630934202.0295820.007884810.30834094.43794302.0280150.003486640.20457754.304321502.0272140.001251940.12244614.237321602.0270760.000869010.10199764.225757705.10 × 10−54.109339001.00015801.000521Table 3
E (X), Var (X), Skew (Y), and Kur (Y) of the RW model.
a3E (Y)Var (Y)Skew (Y)Kur (Y)4.011.2245880.26851075.5650775436.5964.11.2173870.24911105.236318529.59174.251.2063320.22121624.790364204.83094.51.1901510.18425634.23883798.8014551.164230.13376143.53507248.0915161.1287870.07995782.80556624.6781271.1057670.05327192.42509717.5340281.0896520.03807442.1892714.16589101.0686290.02226241.91033910.97857121.0555470.01460851.7498099.461719151.0431670.00885811.6052458.282494201.0314530.004732761.4738847.333494251.0247320.002940271.4004866.852401301.0203740.002002721.3535666.562309351.0173210.001451511.3209686.368528401.0150630.001100171.2969986.229997451.0133260.000862541.2786286.126068501.0119470.000694361.2640996.045237601.0098990.000477791.2425795.927632751.0078740.000303011.2213745.814377801.0073720.000265721.2161225.786696901.0065370.000209161.2074105.7410881001.0058720.000168921.2004785.7051761011.0058130.000165541.1998625.7019951021.0057550.000162271.1992585.69888Thepth incomplete moment, say Ip,Yt, of the RV Y can be derived from (9) as(10)Ip,Yt=∫−∞typfV¯ydy=∑κ4=0∞ϒκ4∫−∞typπ1+κ4y;a3dy,then(11)Ip,Yt|a3>p=∑κ4=0∞ϒκ41+κ4p/a3γ1−pa3,1+κ41ta3,where γ∇,p is the incomplete gamma function.(12)γ∇,p|∇≠0,−1,−2,…=∫0pt∇−1exp−tdt=∑κ4=0∞p∇+κ4κ4!∇+κ4.The first incomplete moment given by (11) with p=1 is(13)I1,Yt|a3>1=∑κ4=0∞wκ41+κ4a3γ1−1a3,1+κ41ta3.


## 2.3. Moment Generating Function (MGF)

The MGF ofY can be derived from equation (7) as MYt=∑κ4=0∞ϒκ4MY,1+κ4t;a3, where MY,1+κ4t;a3 is the MGF of the RW model with scale parameter 1+κ4a3 and shape parameter a3, then(14)MYt|a3>p=Γ1−pa3∑κ4=0∞∑p=0∞tpp!ϒκ41+κ4p/a3.


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

Themth moment of the RLf can be obtained by using lm,Yt=EY−tm|Y>t,m∈ℕ or using lm,Yt=1/1−FV¯t∫t∞Y−tmdFV¯y. Therefore, the RLf can be written as(15)lm,Yt=Γ1−m/a3,1+κ41/ta31−FV¯t∑κ4=0∞cκ411+κ4m/a3,∀a3>m,where cκ41=ϒκ4∑p=0mmp−tm−p,Γ∇,p|y>0=∫p∞t∇−1exp−tdt, and Γ∇,p+γ∇,p=Γ∇. Analogously, the mth moment of RRLf simply can be obtained from Lm,Yt=Et−Ym|Y≤t,t>0andm∈ℕ or from Lm,Yt=1/FV¯t∫0tt−YmdFV¯y. Then, the mth moment of the RRLf of Y becomes(16)Lm,Yt=γ1−m/a3,1+κ41/ta3FV¯t∑κ4=0∞cκ421+κ4m/a3,∀a3>m,where cκ42=ϒκ4∑p=0m−1pmptm−p.


## 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

Lety1,y2,…,ym be an observed random sample (ORS) from size q from the GOLLRW distribution with parameters a1,a2, and a3. Let V¯ be the 3×1 vector of parameters. For obtaining the MLEs of a1,a2, and a3, we derive the below log-likelihood function (LLF).(17)ℓ=ℓV¯=qloga1a2a3−a3+1∑i=1qlogyi:q−a1a2∑i=1qyi:q−a3+2∑i=1qlogexp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1+a1−1∑i=1qlog1−exp−a2yi:q−a3.The score vector is available if needed and can be computed numerically.


### 3.2. The CVME

The CVME of the parametersa1,a2, and a3 are obtained via minimizing the function CVMV¯, where(18)CVMV¯=112q−1+∑i=1qFV¯yi:q−εi,q12,with respect to (WRT), the parameters a1,a2,anda3, respectively, where εi,q1=2i−1/2q and(19)CVMV¯=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q12.The CVME of the parametersa1,a2, and a3 are derived by resolving(20)∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q1∇a1yi:q,V¯=0,∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q1∇a2yi:q,V¯=0,∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q1∇a3yi:q,V¯=0,where ∇a1yi:q,V¯=∂FV¯yi:q/∂a1,∇a2yi:q,V¯=∂FV¯yi:q/∂a2, and ∇a3yi:q,V¯=∂FV¯yi:q/∂a3.


### 3.3. The OLSE Method

LetFV¯yi:q denote the CDF of GOLLRW model and let y1<y2<⋯<yq be the q ordered ORS. The OLSEs are derived by minimizing(21)OLSEV¯=∑i=1qFV¯yi:q−εi,q22;then, we have(22)OLSEV¯=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q22,where εi,q2=i/q+1. The LSEs are derived by solving(23)0=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a1yi:q,V¯,0=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a2yi:q,V¯,0=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a3yi:q,V¯,where ∇a1yi:q,V¯,∇a2yi:q,V¯, and ∇a3yi:q,V¯ are defined.


### 3.4. The WLSE Method

The WLSE can be gotten via minimizingWLSEV¯ WRT a1,a2, and a3, where(24)WLSEV¯=∑i=1qεi,q3FV¯yi:q−εi,q22,where εi,q3=q+122+q/i1−i+q. The WLSEs are obtained by solving(25)0=∑i=1qεi,q3exp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a1yi:q,V¯,0=∑i=1qεi,q3exp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a2yi:q,V¯,0=∑i=1qεi,q3exp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a3yi:q,V¯,where ∇a1yi:q,V¯,∇a2yi:q,V¯, and ∇a3yi:q,V¯ are defined.


### 3.5. The ADE Method

The ADE ofa^1ADE,a^2ADE, and a^3ADE are derived from minimizing(26)ADEV¯=−q−q−1∑i=1q2i−1logFV¯yi:q+log1−FV¯y−i+1+q:q,i.e., the parameter estimates of a^1ADE,a^2ADE, and a^3ADE follow by solving the following nonlinear system:(27)∂∂a1ADEV¯=∂∂a2ADEV¯=∂∂a3ADEV¯=0.


### 3.6. The RTADE Method

The RTADE ofa^1RTADE,a^2RTADE, and a^3RTADE are obtained by minimizing(28)RTADEV¯=12q−2∑i=1qFV¯yi:q−1q2i−1log1−FV¯y−i+1+q⋅q.The parameter estimates ofa^1RTADE,a^2RTADE, and a^3RTADE follow by solving the nonlinear equations.(29)∂∂a1RTADEV¯=0,∂∂a2RTADEV¯=0,∂∂a3RTADEV¯=0.


### 3.7. The LTADE Method

The RTADE ofa^1LTADE,a^2LTADE, and a^3LTADE are obtained by minimizing(30)LTADEV¯=−32q+2∑i=1qFV¯yi:q−1q∑i=1q2i−1logFV¯yi:q.The parameter estimates ofa^1LTADE,a^2LTADE, and a^3LTADE follow by solving the nonlinear equations.(31)∂LTADEV¯∂a1=0,∂LTADEV¯∂a2=0,∂LTADEV¯∂a3=0.


## 3.1. The MLE

Lety1,y2,…,ym be an observed random sample (ORS) from size q from the GOLLRW distribution with parameters a1,a2, and a3. Let V¯ be the 3×1 vector of parameters. For obtaining the MLEs of a1,a2, and a3, we derive the below log-likelihood function (LLF).(17)ℓ=ℓV¯=qloga1a2a3−a3+1∑i=1qlogyi:q−a1a2∑i=1qyi:q−a3+2∑i=1qlogexp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1+a1−1∑i=1qlog1−exp−a2yi:q−a3.The score vector is available if needed and can be computed numerically.


## 3.2. The CVME

The CVME of the parametersa1,a2, and a3 are obtained via minimizing the function CVMV¯, where(18)CVMV¯=112q−1+∑i=1qFV¯yi:q−εi,q12,with respect to (WRT), the parameters a1,a2,anda3, respectively, where εi,q1=2i−1/2q and(19)CVMV¯=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q12.The CVME of the parametersa1,a2, and a3 are derived by resolving(20)∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q1∇a1yi:q,V¯=0,∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q1∇a2yi:q,V¯=0,∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q1∇a3yi:q,V¯=0,where ∇a1yi:q,V¯=∂FV¯yi:q/∂a1,∇a2yi:q,V¯=∂FV¯yi:q/∂a2, and ∇a3yi:q,V¯=∂FV¯yi:q/∂a3.


## 3.3. The OLSE Method

LetFV¯yi:q denote the CDF of GOLLRW model and let y1<y2<⋯<yq be the q ordered ORS. The OLSEs are derived by minimizing(21)OLSEV¯=∑i=1qFV¯yi:q−εi,q22;then, we have(22)OLSEV¯=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q22,where εi,q2=i/q+1. The LSEs are derived by solving(23)0=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a1yi:q,V¯,0=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a2yi:q,V¯,0=∑i=1qexp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a3yi:q,V¯,where ∇a1yi:q,V¯,∇a2yi:q,V¯, and ∇a3yi:q,V¯ are defined.


## 3.4. The WLSE Method

The WLSE can be gotten via minimizingWLSEV¯ WRT a1,a2, and a3, where(24)WLSEV¯=∑i=1qεi,q3FV¯yi:q−εi,q22,where εi,q3=q+122+q/i1−i+q. The WLSEs are obtained by solving(25)0=∑i=1qεi,q3exp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a1yi:q,V¯,0=∑i=1qεi,q3exp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a2yi:q,V¯,0=∑i=1qεi,q3exp−a1a2yi:q−a3exp−a1a2yi:q−a3+1−exp−a2yi:q−a3a1−εi,q2∇a3yi:q,V¯,where ∇a1yi:q,V¯,∇a2yi:q,V¯, and ∇a3yi:q,V¯ are defined.


## 3.5. The ADE Method

The ADE ofa^1ADE,a^2ADE, and a^3ADE are derived from minimizing(26)ADEV¯=−q−q−1∑i=1q2i−1logFV¯yi:q+log1−FV¯y−i+1+q:q,i.e., the parameter estimates of a^1ADE,a^2ADE, and a^3ADE follow by solving the following nonlinear system:(27)∂∂a1ADEV¯=∂∂a2ADEV¯=∂∂a3ADEV¯=0.


## 3.6. The RTADE Method

The RTADE ofa^1RTADE,a^2RTADE, and a^3RTADE are obtained by minimizing(28)RTADEV¯=12q−2∑i=1qFV¯yi:q−1q2i−1log1−FV¯y−i+1+q⋅q.The parameter estimates ofa^1RTADE,a^2RTADE, and a^3RTADE follow by solving the nonlinear equations.(29)∂∂a1RTADEV¯=0,∂∂a2RTADEV¯=0,∂∂a3RTADEV¯=0.


## 3.7. The LTADE Method

The RTADE ofa^1LTADE,a^2LTADE, and a^3LTADE are obtained by minimizing(30)LTADEV¯=−32q+2∑i=1qFV¯yi:q−1q∑i=1q2i−1logFV¯yi:q.The parameter estimates ofa^1LTADE,a^2LTADE, and a^3LTADE follow by solving the nonlinear equations.(31)∂LTADEV¯∂a1=0,∂LTADEV¯∂a2=0,∂LTADEV¯∂a3=0.


## 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 onN = 1000 data sets generated from the new model where n=50,100,300, and 500, where(32)ScenarioI2.00.60.8ScenarioII0.51.52.0ScenarioIII0.90.90.9Figure2 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 BIASV¯; root mean − standard error RMSEV¯; 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 BIASV¯ tends to 0 as n increases which shows that all estimators are “nonbiased.”(ii)
The RMSEV¯ tends to 0 as n increases which shows the incidence of “consistency property.”Figure 2
Density functions for three scenarios.Table 4
Assessment results for parametersa1 = 2.0, a2 = 0.6, and a3 = 0.8.
nBIASa1BIASa2BIASa3RMSEa1RMSEa2RMSEa3AD-absAD-maxMLE500.021250.002410.008830.240950.050200.084510.003770.00718CVM−0.033610.005900.001960.283830.056120113110.004790.00800OLS−0.040760.015960.019880.284220.056810.127730.014030.02079WLS0.190870.025510.077920.379260.060050.135780.036720.07189ADE−0.058620.00436−0.011590.259280.052330.097130.006670.01160RTADE0.006990.00511−0.014930.348090.057050.110650.003120.00573LTADE−0.001890.002560.006430.332870.052060.119020.002680.00480MLE1000.014650.001730.006950.169130.036050.059490.003360.00655CVM−0.011770.00004−0.000950.200470.036800.081390.001060.00166OLS−0.008260.007560.013340.194270.038200.084280.007190.01191WLS0.138000.014320.053970.244640.040620.088160.024800.04815ADE−0.019140.00186−0.002940.176130.036210.066030.002100.00392RTADE−0.005840.00364−0.007890.224400.038580.076940.002530.00472LTADE0.008130.001520.004770.230520.035900.078340.002140.00418MLE3000.006050.001530.003120.098020.020400.034230.001850.00353CVM−0.003740.000660.001150.114410.021370.048730.000630.00090OLS−0.004440.001980.003130.113680.021780.048810.001860.00286WLS0.063550.006460.024140.128110.022950.045330.011430.02215ADE−0.008650.00014−0.001910.102560.021160.039630.001010.00162RTADE0.003480.00148−0.000170.133450.022150.043710.001150.00186LTADE−0.00102−0.000160.000250.137370.020870.045730.000100.00017MLE5000.004330.000560.001590.078580.016090.027190.000830.00162CVM−0.003660.000630.000490.084630.016200.035690.000540.00084OLS−0.000520.002190.003790.086100.016240.035370.002110.00362WLS0.050270.005490.019240.099100.017460.034820.009250.01797ADE−0.002360.001060.000560.078250.015760.029090.000890.00130RTADE0.002560.00051−0.000490.104150.017940.034600.000360.00061LTADE0.007070.000990.02710.101650.015610.033870.001410.00275Table 5
Assessment results for parametersa1 = 0.5, a2 = 1.5, and a3 = 2.0.
nBIASa1BIASa2BIASa3RMSEa1RMSEa2RMSEa3AD-absAD-maxMLE500.008850.062170.039640.062690.295590.209360.007930.01690CVM−0.009080.08121−0.013760.071190.458810.306630.008210.01095OLS−0.003280.18501−0.042160.073880.517600.317440.018750.02491WLS0.052830.241340.177740.093880.461300.310770.034700.07045ADE−0.009690.05600−0.044300.065600.369560.241700.007330.01349RTADE−0.000630.125710.010590.084640.554540.349660.011470.01982LTADE0.003760.04841−0.025870.080200.322950.235330.005610.00812MLE1000.005800.032310.023560.042070.198780.138360.004650.00970CVM−0.007060.02614−0.007570.048470.301430.220600.002840.00563OLS−0.002720.09558−0.022140.047480.327480.229130.009900.01285WLS0.033060.138940.117020.059230.268190.200830.022270.04464ADE−0.005160.03187−0.022960.043960.249230.168990.004010.00724RTADE−0.000450.050010.006900.057500.353800.235520.004600.00821LTADE−0.000020.01576−0.016930.059380.226350.167500.002150.00332MLE3000.001100.010300.005100.023630.107390.076800.001160.00256CVM−0.002640.00716−0.002650.027470.152470.129390.000930.00192OLS−0.001310.01964−0.004930.027570.159430.125870.002090.00262WLS0.015830.053740.057830.031990.131470.106050.010520.02035ADE−0.002290.00008−0.007460.024770.133050.092430.001240.00201RTADE−0.000250.019140.000870.032600.182490.134690.001840.00301LTADE−0.000200.00279−0.003940.033690.124920.094570.000450.00078MLE500−0.000990.00209−0.001180.018750.084580.060020.000360.00072CVM−0.000410.007040.000640.021670.117370.098420.000670.00100OLS−0.000910.01876−0.006960.021120.124280.097960.002070.00266WLS−0.011520.044600.042260.024040.101590.079500.007940.01575ADE−0.001550.00665−0.007070.019130.104620.071970.001050.00201RTADE−0.001860.00904−0.000410.024440.140300.101290.001030.00190LTADE0.000270.00469−0.002890.025880.095250.072260.000560.00076Table 6
Assessment results for parametersa1 = 0.9, a2 = 0.9, and a3 = 0.9.
nBIASa1BIASa2BIASa3RMSEa1RMSEa2RMSEa3AD-absAD-maxMLE500.013810.015590.017700.111280.143100.098040.006410.01246CVM−0.014440.01793−0.010150.126250.167440.129310.005280.00981OLS−0.014520.06077−0.021990.129610.187790.135640.015800.02358WLS0.084940.091610.079970.169290.188040.150790.032920.06504ADE−0.025880.01931−0.026400.115380.157440.108960.009230.01718RTADE0.006900.025300.011320.149650.187400.159600.006420.01257LTADE0.008510.015890.001070.145770.143760.115570.004150.00779MLE1000.005700.015420.006610.078420.098850.067040.004030.00801CVM−0.003800.01396−0.03680.086600.117610.088790.003660.00530OLS−0.009410.02469−0.012170.090510.121810.091180.006660.01095WLS0.054250.051550.051240.106950.119680.092980.020760.04056ADE−0.013740.00448−0.013490.082980.107950.075540.004570.00786RTADE−0.001280.017050.001230.104640.135740.111870.004240.00621LTADE0.002990.00385−0.000030.106000.099970.082420.001070.00206MLE3000.001830.004070.001750.043420.055140.036750.001070.00211CVM−0.002350.00407−0.001820.051710.066600.053920.001110.00192OLS−0.006450.00625−0.006080.049180.063690.051180.002340.00443WLS0.027450.022200.026270.057930.062260.049970.010350.01981ADE−0.00733−0.00022−0.006410.045310.058580.042220.002310.00352RTADE−0.002950.00490−0.001880.059410.073300.062090.001320.00225LTADE0.002600.004910.000650.059700.056570.046200.001290.00247MLE5000.001430.001780.001740.032930.043480.027470.000730.00144CVM−0.001500.00163−0.000960.038750.050720.040040.000500.00095OLS−0.001500.00521−0.002100.038750.048980.039770.001400.00214WLS0.020260.017200.019270.043380.048190.037090.007730.01493ADE−0.003170.00139−0.003170.035430.045490.032510.001060.00189RTADE0.001200.003450.001410.048290.058240.050540.000900.00177LTADE0.000670.00138−0.000090.049030.045580.037900.000360.00064Generally, 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) ∀n=50,100,300,500 is (0.24095, 0.16913, 0.09802, 0.07858). RMSE (a2) ∀n=50,100,300,500 is (0.05020, 0.03605, 0.02040, 0.01609). RMSE (a3) ∀n=50,100,300,500 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) ∀n=50,100,300,500 is (0.06269, 0.04207, 0.02363, 0.01875). RMSE (a2) ∀n=50,100,300,500 is (0.29559, 0.19878, 0.10739, 0.08458). RMSE (a3) ∀n=50,100,300,500 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) ∀n=50,100,300,500 is (0.11128, 0.07842, 0.04342, 0.03293). RMSE (a2) ∀n=50,100,300,500 is (0.14310, 0.09885, 0.05514, 0.04348). RMSE (a3) ∀n=50,100,300,500 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 whenn 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 (C1); Anderson-Darling (C2); and Kolmogorov–Smirnov test (KS-test) and its corresponding p value (P.V). The new model is compared with several common competitive models; Table 7 lists the competitive models and their corresponding abbreviations.Table 7
The competitive models.
Competitive modelAbbreviationKumaraswamy-RWK-RWExponentiated-RWE-RWTransmuted-RWT-RWBeta-RWB-RWMarshal-Olkin-RWMO-RWOdd-log-logistic-ERWOLL-ERWGeneralized OLL-RRGOLL-RROdd-log-logistic-EIROLL-EIRMcDonald-RWMc-RWOdd-log-logistic-RROLL-RRReciprocal WeibullRW


#### 5.1.1. Breaking Stress Data

The 1st 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.Figure 3
TTT plot, box plot, Q-Q plot, and KDE for stress data set.Figure 4
P-P plot, EPF, ECDF, and EHRF for stress data set.The statisticsC1, C2 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 C1, C2, 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.Table 8
C₁, C₂, KS-test, and P.V for stress data set.
ModelGoodness-of-fit (GOF) criteriaC1C2KS-testP.VGOLLRW0.06230.48350.06370.8123OLLE-RW0.12030.96390.5561<0.0001OLLE-RR0.15531.21200.6550<0.0001OLL-RR0.15531.21200.6550<0.0001RW0.10900.76570.08740.4282K-RW0.08120.62170.07590.6118E-RW0.10910.76580.08740.4287B-RW0.08090.62070.07570.6147T-RW0.08710.62090.07820.5734MO-RW0.08860.61420.07630.5168Mc-RW0.13331.06080.08070.5332Table 9
MLEs (SEs) for stress data set.
ModelMLEsa1a2caa3GOLLRW1.70152.36042.8549(0.6555)(0.9817)(1.0017)OLLE-RW0.13513.72160.929621.319(0.011)(0.0034)(0.0033)(0.0034)OLLE-RR0.49460.0671.74262(0.0414)(0.7195)(9.3007)OLL-RR0.494590.452420.041350.03869RW1.39684.3724(0.0336)(0.3278)K-RW0.84891.62391.63413.4208(16.083)(0.6979)(9.049)(0.7635)E-RW0.93951.41690.9395(3.543)(2.568)(0.3278)B-RW0.73461.58301.66843.5112(1.5290)(0.7132)(0.7662)(0.9683)T-RW−0.71661.26564.7121(0.2616)(0.0579)(0.3657)MO-RW0.00336.22961.2419(0.0009)(1.0134)(0.1181)Mc-RW0.850344.42319.8590.020346.974(0.1353)(25.100)(6.706)(0.0060)(21.871)


#### 5.1.2. Glass Fiber Data

The 2nd 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 C1, C2, 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 C1, C2, and KS-test, and the biggest value of the P.V; 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.Figure 5
TTT plot, box plot, Q-Q plot, and KDE for fiber data set.Figure 6
P-P plot, EPF, ECDF, and EHRF for fiber data set.Table 10
C₁, C₂, KS-test, and P.V for fiber data set.
ModelGOF criteriaC1C2KS-testP.VGOLLRW0.05080.41160.069110.9039GOLL-RR0.05250.45300.070330.8926OLLE-RW0.10490.83250.55196<0.0001OLLE-RR0.15021.146970.67949<0.0001OLL-RR0.15021.146970.67951<0.0001RW0.07070.53320.07720.8185K-RW0.06340.49810.07150.8810E-RW0.07070.53320.07720.8187B-RW0.06400.50080.07160.8804T-RW0.06550.49390.07350.8470MO-RW0.06290.49020.08130.7685Mc-RW0.11610.91930.08310.7455Table 11
MLEs (SEs) for fiber data set.
ModelMLEsGOLLRW1.47293.68933.9993(0.6315)(2.2764)(1.5278)OLLE-RW0.14490.008791.299724.878(0.0129)(0.000)(0.000)(0.000)GOLL-RR3.0322.15680.86599(0.325)(9.145)(1.836)OLLE-RR0.50250.07161.7048(0.0529)(1.13062)(13.47)OLL-RR0.502510.455990.0529460.04865RW1.41085.4377(0.0344)(0.5192)K-RW0.28551.28241.91424.7731(9.1338)(0.6388)(12.836)(1.3134)E-RW0.90591.43675.4379(2.764)(4.324)(0.5193)B-RW1.29961.26491.39454.7927(4.4378)(0.6640)(0.9304)(1.4641)T-RW0.77781.54914.3139(0.2477)(0.0655)(0.5849)MO-RW0.00235.23831.4537(0.0004)(0.8209)(0.1650)Mc-RW56.22714.9530.007329.104(30.539)(4.733)(0.0013)(11.304)


#### 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 C1, C2, 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 C1, C2, 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.Figure 7
TTT plot, box plot, Q-Q plot, and KDE for relief time data set.Figure 8
P-P plot, EPF, ECDF, and EHRF for relief time data set.Table 12
C1, C2, KS-test, and P.V for relief time data set.
ModelGOF criteriaC1C2KS-testP.VGOLLRW0.103800.81630.10430.6482GOLL-RW0.195511.34980.110080.5797OLLE-RW0.15771.098760.53498<0.0001RW0.32332.03010.15060.2066E-RW0.32332.03010.15060.2064T-RW0.28231.81520.13700.3045Table 13
MLEs (SEs) for relief time data set.
ModelMLEsGOLLRW29.50360.64130.1432(49.1402)(0.0828)(0.2377)GOLL-RW1.9610.1111.4123(0.234)(0.000)(0.000)OLLE-RW0.06690.004590.355832.561(0.0076)(0.0028)(0.0047)(0.006)RW0.48593.2078(0.0227)(0.3263)E-RW0.90470.50133.2077(18.784)(3.2444)(0.3263)T-RW−0.58160.44003.4974(0.2787)(0.0290)(0.3527)


### 5.2. Comparing Estimation Methods under Uncensored Data

For comparing estimation methods via real data applications, we introduce two examples. The comparison is based onC1 and C2. Tables 14–16 give the results for comparing estimation methods using breaking stress data and glass fiber data, respectively.Table 14
Comparing estimation methods using breaking stress data.
Methoda1a2a3C1C2MLE1.7982.2222.7090.060960.478CVM2.2441.8172.2320.058640.473OLS2.2071.8542.2600.058670.474WLS0.9825.6305.0010.137070.927ADE2.1201.8992.3400.058920.473RTADE2.1271.8892.3320.058920.473LTADE2.1341.8852.3200.058910.473Table 15
Comparing estimation methods using glass fiber data.
Methoda1a2a3C1C2MLE1.7412.9253.4290.048660.40446CVM2.5621.9072.4070.049670.42531OLS2.4961.9822.4710.049660.42517WLS1.8362.9983.4580.047450.39247ADE2.2642.1592.7050.048750.41466RTADE2.1042.2972.8700.048600.41149LTADE2.0592.4222.9840.048120.40564Table 16
Values ofρ^j, Uj, C^1j,C^2j,C^3j, and ej.
ρ^j7.515.6531.4566.4Uj6.006.007.0012.0C^1j0.940.461.300.31C^2j0.56−0.46−0.490.20C^3j0.54170.46890.78640.2153ej2.74962.74962.74962.7496


#### 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. Table14 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 C1 = 0.05864 and C2 = 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. Table15 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 C1 = 0.04745 and C2 = 0.39247, however all other methods performed well.


## 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 (C1); Anderson-Darling (C2); and Kolmogorov–Smirnov test (KS-test) and its corresponding p value (P.V). The new model is compared with several common competitive models; Table 7 lists the competitive models and their corresponding abbreviations.Table 7
The competitive models.
Competitive modelAbbreviationKumaraswamy-RWK-RWExponentiated-RWE-RWTransmuted-RWT-RWBeta-RWB-RWMarshal-Olkin-RWMO-RWOdd-log-logistic-ERWOLL-ERWGeneralized OLL-RRGOLL-RROdd-log-logistic-EIROLL-EIRMcDonald-RWMc-RWOdd-log-logistic-RROLL-RRReciprocal WeibullRW


### 5.1.1. Breaking Stress Data

The 1st 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.Figure 3
TTT plot, box plot, Q-Q plot, and KDE for stress data set.Figure 4
P-P plot, EPF, ECDF, and EHRF for stress data set.The statisticsC1, C2 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 C1, C2, 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.Table 8
C₁, C₂, KS-test, and P.V for stress data set.
ModelGoodness-of-fit (GOF) criteriaC1C2KS-testP.VGOLLRW0.06230.48350.06370.8123OLLE-RW0.12030.96390.5561<0.0001OLLE-RR0.15531.21200.6550<0.0001OLL-RR0.15531.21200.6550<0.0001RW0.10900.76570.08740.4282K-RW0.08120.62170.07590.6118E-RW0.10910.76580.08740.4287B-RW0.08090.62070.07570.6147T-RW0.08710.62090.07820.5734MO-RW0.08860.61420.07630.5168Mc-RW0.13331.06080.08070.5332Table 9
MLEs (SEs) for stress data set.
ModelMLEsa1a2caa3GOLLRW1.70152.36042.8549(0.6555)(0.9817)(1.0017)OLLE-RW0.13513.72160.929621.319(0.011)(0.0034)(0.0033)(0.0034)OLLE-RR0.49460.0671.74262(0.0414)(0.7195)(9.3007)OLL-RR0.494590.452420.041350.03869RW1.39684.3724(0.0336)(0.3278)K-RW0.84891.62391.63413.4208(16.083)(0.6979)(9.049)(0.7635)E-RW0.93951.41690.9395(3.543)(2.568)(0.3278)B-RW0.73461.58301.66843.5112(1.5290)(0.7132)(0.7662)(0.9683)T-RW−0.71661.26564.7121(0.2616)(0.0579)(0.3657)MO-RW0.00336.22961.2419(0.0009)(1.0134)(0.1181)Mc-RW0.850344.42319.8590.020346.974(0.1353)(25.100)(6.706)(0.0060)(21.871)


### 5.1.2. Glass Fiber Data

The 2nd 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 C1, C2, 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 C1, C2, and KS-test, and the biggest value of the P.V; 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.Figure 5
TTT plot, box plot, Q-Q plot, and KDE for fiber data set.Figure 6
P-P plot, EPF, ECDF, and EHRF for fiber data set.Table 10
C₁, C₂, KS-test, and P.V for fiber data set.
ModelGOF criteriaC1C2KS-testP.VGOLLRW0.05080.41160.069110.9039GOLL-RR0.05250.45300.070330.8926OLLE-RW0.10490.83250.55196<0.0001OLLE-RR0.15021.146970.67949<0.0001OLL-RR0.15021.146970.67951<0.0001RW0.07070.53320.07720.8185K-RW0.06340.49810.07150.8810E-RW0.07070.53320.07720.8187B-RW0.06400.50080.07160.8804T-RW0.06550.49390.07350.8470MO-RW0.06290.49020.08130.7685Mc-RW0.11610.91930.08310.7455Table 11
MLEs (SEs) for fiber data set.
ModelMLEsGOLLRW1.47293.68933.9993(0.6315)(2.2764)(1.5278)OLLE-RW0.14490.008791.299724.878(0.0129)(0.000)(0.000)(0.000)GOLL-RR3.0322.15680.86599(0.325)(9.145)(1.836)OLLE-RR0.50250.07161.7048(0.0529)(1.13062)(13.47)OLL-RR0.502510.455990.0529460.04865RW1.41085.4377(0.0344)(0.5192)K-RW0.28551.28241.91424.7731(9.1338)(0.6388)(12.836)(1.3134)E-RW0.90591.43675.4379(2.764)(4.324)(0.5193)B-RW1.29961.26491.39454.7927(4.4378)(0.6640)(0.9304)(1.4641)T-RW0.77781.54914.3139(0.2477)(0.0655)(0.5849)MO-RW0.00235.23831.4537(0.0004)(0.8209)(0.1650)Mc-RW56.22714.9530.007329.104(30.539)(4.733)(0.0013)(11.304)


### 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 C1, C2, 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 C1, C2, 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.Figure 7
TTT plot, box plot, Q-Q plot, and KDE for relief time data set.Figure 8
P-P plot, EPF, ECDF, and EHRF for relief time data set.Table 12
C1, C2, KS-test, and P.V for relief time data set.
ModelGOF criteriaC1C2KS-testP.VGOLLRW0.103800.81630.10430.6482GOLL-RW0.195511.34980.110080.5797OLLE-RW0.15771.098760.53498<0.0001RW0.32332.03010.15060.2066E-RW0.32332.03010.15060.2064T-RW0.28231.81520.13700.3045Table 13
MLEs (SEs) for relief time data set.
ModelMLEsGOLLRW29.50360.64130.1432(49.1402)(0.0828)(0.2377)GOLL-RW1.9610.1111.4123(0.234)(0.000)(0.000)OLLE-RW0.06690.004590.355832.561(0.0076)(0.0028)(0.0047)(0.006)RW0.48593.2078(0.0227)(0.3263)E-RW0.90470.50133.2077(18.784)(3.2444)(0.3263)T-RW−0.58160.44003.4974(0.2787)(0.0290)(0.3527)


## 5.1.1. Breaking Stress Data

The 1st 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.Figure 3
TTT plot, box plot, Q-Q plot, and KDE for stress data set.Figure 4
P-P plot, EPF, ECDF, and EHRF for stress data set.The statisticsC1, C2 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 C1, C2, 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.Table 8
C₁, C₂, KS-test, and P.V for stress data set.
ModelGoodness-of-fit (GOF) criteriaC1C2KS-testP.VGOLLRW0.06230.48350.06370.8123OLLE-RW0.12030.96390.5561<0.0001OLLE-RR0.15531.21200.6550<0.0001OLL-RR0.15531.21200.6550<0.0001RW0.10900.76570.08740.4282K-RW0.08120.62170.07590.6118E-RW0.10910.76580.08740.4287B-RW0.08090.62070.07570.6147T-RW0.08710.62090.07820.5734MO-RW0.08860.61420.07630.5168Mc-RW0.13331.06080.08070.5332Table 9
MLEs (SEs) for stress data set.
ModelMLEsa1a2caa3GOLLRW1.70152.36042.8549(0.6555)(0.9817)(1.0017)OLLE-RW0.13513.72160.929621.319(0.011)(0.0034)(0.0033)(0.0034)OLLE-RR0.49460.0671.74262(0.0414)(0.7195)(9.3007)OLL-RR0.494590.452420.041350.03869RW1.39684.3724(0.0336)(0.3278)K-RW0.84891.62391.63413.4208(16.083)(0.6979)(9.049)(0.7635)E-RW0.93951.41690.9395(3.543)(2.568)(0.3278)B-RW0.73461.58301.66843.5112(1.5290)(0.7132)(0.7662)(0.9683)T-RW−0.71661.26564.7121(0.2616)(0.0579)(0.3657)MO-RW0.00336.22961.2419(0.0009)(1.0134)(0.1181)Mc-RW0.850344.42319.8590.020346.974(0.1353)(25.100)(6.706)(0.0060)(21.871)


## 5.1.2. Glass Fiber Data

The 2nd 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 C1, C2, 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 C1, C2, and KS-test, and the biggest value of the P.V; 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.Figure 5
TTT plot, box plot, Q-Q plot, and KDE for fiber data set.Figure 6
P-P plot, EPF, ECDF, and EHRF for fiber data set.Table 10
C₁, C₂, KS-test, and P.V for fiber data set.
ModelGOF criteriaC1C2KS-testP.VGOLLRW0.05080.41160.069110.9039GOLL-RR0.05250.45300.070330.8926OLLE-RW0.10490.83250.55196<0.0001OLLE-RR0.15021.146970.67949<0.0001OLL-RR0.15021.146970.67951<0.0001RW0.07070.53320.07720.8185K-RW0.06340.49810.07150.8810E-RW0.07070.53320.07720.8187B-RW0.06400.50080.07160.8804T-RW0.06550.49390.07350.8470MO-RW0.06290.49020.08130.7685Mc-RW0.11610.91930.08310.7455Table 11
MLEs (SEs) for fiber data set.
ModelMLEsGOLLRW1.47293.68933.9993(0.6315)(2.2764)(1.5278)OLLE-RW0.14490.008791.299724.878(0.0129)(0.000)(0.000)(0.000)GOLL-RR3.0322.15680.86599(0.325)(9.145)(1.836)OLLE-RR0.50250.07161.7048(0.0529)(1.13062)(13.47)OLL-RR0.502510.455990.0529460.04865RW1.41085.4377(0.0344)(0.5192)K-RW0.28551.28241.91424.7731(9.1338)(0.6388)(12.836)(1.3134)E-RW0.90591.43675.4379(2.764)(4.324)(0.5193)B-RW1.29961.26491.39454.7927(4.4378)(0.6640)(0.9304)(1.4641)T-RW0.77781.54914.3139(0.2477)(0.0655)(0.5849)MO-RW0.00235.23831.4537(0.0004)(0.8209)(0.1650)Mc-RW56.22714.9530.007329.104(30.539)(4.733)(0.0013)(11.304)


## 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 C1, C2, 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 C1, C2, 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.Figure 7
TTT plot, box plot, Q-Q plot, and KDE for relief time data set.Figure 8
P-P plot, EPF, ECDF, and EHRF for relief time data set.Table 12
C1, C2, KS-test, and P.V for relief time data set.
ModelGOF criteriaC1C2KS-testP.VGOLLRW0.103800.81630.10430.6482GOLL-RW0.195511.34980.110080.5797OLLE-RW0.15771.098760.53498<0.0001RW0.32332.03010.15060.2066E-RW0.32332.03010.15060.2064T-RW0.28231.81520.13700.3045Table 13
MLEs (SEs) for relief time data set.
ModelMLEsGOLLRW29.50360.64130.1432(49.1402)(0.0828)(0.2377)GOLL-RW1.9610.1111.4123(0.234)(0.000)(0.000)OLLE-RW0.06690.004590.355832.561(0.0076)(0.0028)(0.0047)(0.006)RW0.48593.2078(0.0227)(0.3263)E-RW0.90470.50133.2077(18.784)(3.2444)(0.3263)T-RW−0.58160.44003.4974(0.2787)(0.0290)(0.3527)


## 5.2. Comparing Estimation Methods under Uncensored Data

For comparing estimation methods via real data applications, we introduce two examples. The comparison is based onC1 and C2. Tables 14–16 give the results for comparing estimation methods using breaking stress data and glass fiber data, respectively.Table 14
Comparing estimation methods using breaking stress data.
Methoda1a2a3C1C2MLE1.7982.2222.7090.060960.478CVM2.2441.8172.2320.058640.473OLS2.2071.8542.2600.058670.474WLS0.9825.6305.0010.137070.927ADE2.1201.8992.3400.058920.473RTADE2.1271.8892.3320.058920.473LTADE2.1341.8852.3200.058910.473Table 15
Comparing estimation methods using glass fiber data.
Methoda1a2a3C1C2MLE1.7412.9253.4290.048660.40446CVM2.5621.9072.4070.049670.42531OLS2.4961.9822.4710.049660.42517WLS1.8362.9983.4580.047450.39247ADE2.2642.1592.7050.048750.41466RTADE2.1042.2972.8700.048600.41149LTADE2.0592.4222.9840.048120.40564Table 16
Values ofρ^j, Uj, C^1j,C^2j,C^3j, and ej.
ρ^j7.515.6531.4566.4Uj6.006.007.0012.0C^1j0.940.461.300.31C^2j0.56−0.46−0.490.20C^3j0.54170.46890.78640.2153ej2.74962.74962.74962.7496


### 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. Table14 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 C1 = 0.05864 and C2 = 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. Table15 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 C1 = 0.04745 and C2 = 0.39247, however all other methods performed well.


## 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. Table14 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 C1 = 0.05864 and C2 = 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. Table15 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 C1 = 0.04745 and C2 = 0.39247, however all other methods performed well.


## 6. Censored Validation and Real Data Analysis

The statistic testTn2 is defined by(33)Tn2=∑j=1n1UUj−ej2+Q,where ej is the number of expected failures (NEF) in the grouped intervals and Uj is the number of observed failures (NOF) in grouping intervals where(34)Q=WTG^−W,W^=C^A^−1Z=W^1,…,W^sT,G^=g^ℒℒ's×s,Wℒ=∑j=1rC^ℒjA^j−1Zj,Zj=1nUj−ej,g^ℒℒ=ı^ℒℒ'−∑j=1rC^ℒjC^ℒ′jA^j−1,j=1,2,…,r=1,2,…,n,ℒ,ℒ′=1,2,…,s.The elements ofC^ are defined by(35)C^ℒj=1n∑i:xi∈IjnΔi∂∂V¯^ℒlnHV¯^yi,where HV¯^yi refers to the cumulative HRF (CHRF) of the GOLLRW distribution. The quadratic form of the modified test statistic can be written as(36)Tn2V¯=∑j=1r1UjUj−ej2+W^Tı^ll′−∑j=1rC^ljC^l′jA^j−1−1W^,where matrices W^, C^, and I^ 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 V^=a^1,a^2,a^3T=1.6385,1.6230,1.2865T, where the data are grouped into r=4 intervals. Table 16 gives the necessary calculus.Second, we calculate the value of theTn2:(37)Tn2=Q+χ2=2.6346+4.2234=6.858.Forε=%5, the critical value χ52=9,4877 which is larger than Tn2=6.858, 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. TheSkewGOLLRWY∈−26.23,2.531, whereas the SkewRWY∈1.199,5.565. Further, the spread for the KurGOLLRWY is ranging from nearly 1.00 to nearly 10810.2, whereas the spread for the KurGOLLRWY only varies from 5.699 to 5436.5. SkewGOLLRWY can be “negative” or “positive,” however the SkewRWY 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]).



---

*Source: 1910909-2022-08-22.xml*