# Marshall–Olkin Extended Gumbel Type-II Distribution: Properties and Applications

Authors: Farwa Willayat; Naz Saud; Muhammad Ijaz; Anita Silvianita; Mahmoud El-Morshedy
Journal: Complexity
 (2022)

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## Abstract

Due to the advance computer technology, the use of probability distributions has been raised up to solve the real life problems. These applications are found in reliability engineering, computer sciences, economics, psychology, survival analysis, and some others. This study offers a new probability model called Marshall–Olkin Extended Gumbel Type-II (MOEGT-II) which can model various shapes of the failure rate function. The proposed distribution is capable to model increasing, decreasing, reverse J-shaped, and upside down bathtub shapes of the failure rate function. Various statistical properties of the proposed distribution are derived such as alternate expressions for the density and distribution function, special cases of MOEGT-II distribution, quantile function, Lorenz curve, and Bonferroni curve. Estimation of the unknown parameters is carried out by the method of maximum likelihood. A simulation study is conducted using three different iterative methods with different samples of sizesn. The usefulness and potentiality of the MOEGT-II distribution have been shown using three real life data sets. The MOEGT-II distribution has been demonstrated as better fit than Exponentiated Gumbel Type-II (EGT-II), Marshall–Olkin Gumbel Type-II (MOGT-II), Gumbel Type-II (GT-II), Marshall–Olkin–Frechet (MOF), Frechet (F), Burr III, Log Logistic (LL), Beta Inverse Weibull (BIW), and Kumaraswamy Inverse Weibull (KIW) distributions.


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## Body

## 1. Introduction

Probability distributions are mostly used in survival analysis for modeling data because it provides insight interest in the nature of various parameters and functions mainly in the failure rate function. The lifetime distributions are used in reliability engineering and life data analysis. For example, Aksoy [1] used the Gamma distribution to model the rainfall data, the Weibull distribution has been used by Keshavan et al. [2] to model fracture strength of glass data, Al-Hasan and Nigmatullin [3] worked out the analysis of wind speed data, and environmental radioactivity data were analyzed by Dahm et al. [4] since the real data set values have different structures and we cannot model these complex data sets with the existing distributions. The existing verities of models can only be applied to the data that have monotonic increasing or decreasing failure function. Surprisingly, this gap has been covered by many researchers working in distribution theory. In this way, the exponential distribution has been modified by many researchers; for example, Gupta and Kundu [5, 6] defined new families of exponential distribution; exponentiated generalized class of distributions was introduced by Cordeiro et al. [7]; and Exponentiated Generalized Gumbel distribution was proposed by Andrade et al. [8].A family of distribution was introduced by Marshall and Olkin [9] by developing a new method for adding a parameter to the existing probability distributions. Special cases of this family were discussed by using Exponential, Weibull, Gamma, and Log Normal distribution. The survival function of a two parametric exponential distribution was obtained by using the survival function of one-parametric exponential distribution. As a result, it was found that the new extended family might seem as much better competitor than the gamma and Weibull families; likewise, two-parametric Weibull distribution was used to derive the survival function of three-parametric Weibull distribution.The cumulative distribution function (cdf) of Marshall–Olkin family is given by(1)GXx=Fx1−1−αF¯x,−∞<x<∞,0<α<∞,where F(x) is the cdf of a continuous type distribution and “α ” is shape parameter. The corresponding probability density function (pdf) of Marshall–Olkin family is(2)gx=αfx1−1−αF¯x2,−∞<x<∞,0<α<∞.In recent years, many researchers have used Marshall–Olkin technique to modify different existing models. For example, Marshall–Olkin Bivariate semipareto (MO-BSP) distribution and Marshall–Olkin Bivariate Pareto (MO-BP) distribution were proposed by Alice and Jose [10]. A three-parameter lifetime distribution named as Marshall–Olkin extended Weibull distribution was introduced by Ghitany et al. [11]. Ghitany et al. [12] derived a new probability distribution to model life time data by extending the Lomax distribution, and this new distribution is called as the Marshall–Olkin Extended Lomax distribution. A four-parameter continuous distribution named as Marshall–Olkin q Weibull distribution and max-min processes was proposed by Jose et al. [13]. Jose [14] presented a new Marshall–Olkin Extended Uniform distribution by using the Marshall–Olkin model on Extended Uniform distribution. The Marshall–Olkin Extended Lindley distribution was introduced by Ghitany et al. [15]; The Marshall–Olkin–Frechet Distribution was proposed by Krishna et al. [16]. Al-Saiari et al. [17] introduced a new three-parameter distribution named as Marshall–Olkin Extended Burr Type XII Distribution.A four-parameter distribution named as a Marshall–Olkin Exponential Weibull (MOEW) distribution for skewed positive data was proposed by Pogány et al. [18]. Saboor and Pogány [19] suggested a new distribution named as the Marshall–Olkin Gamma Weibull (MOGW) distribution. Marshall–Olkin Extended Weibull Distribution was proposed by Ahmad et al. [20]. Marshall–Olkin additive Weibull distribution was proposed by Afify et al. [21]. Exponentiated Gumbel type-II distributed was discovered by Biswas and Gupta [22]. Gillariose et al. [23] introduced a new Marshall–Olkin Modified Lindley Distribution; Marshall Olkin Power Lomax distribution was proposed by Haq et al. [24]; Ogunde et al. [25] proposed the Extended Gumbel Type-II distribution; and Almetwally et al. [26] derived various properties of Marshall–Olkin Alpha Power Weibull distribution.The Exponentiated Generalized Gumbel distribution was proposed by Andrade et al. [8]. Okorie et al. [27] introduced the Kumaraswamy G Exponentiated Gumbel type-II distribution.The generalized form of the standard Gumbel type-II distribution was derived by Okorie et al. [28] called Exponentiated Gumbel type-II (EGT-II). The cdf, survival function, and pdf are(3)FXx=1−1−exp−λx−ρθ,x>0,λ,ρ,θ>0,(4)SXx=1−exp−λx−ρθ,x>0,λ,ρ,θ>0,(5)fXx=θρλx−ρ−1exp−λx−ρ1−exp−λx−ρθ−1,x>0,λ,ρ,θ>0,where “λ” is the scale parameter and “ρ” and “θ” are shape parameters.The Exponentiated Gumbel Type-II Distribution is used for modeling complex data setsHowever, a Gumbel Type-II distribution cannot handle the analysis of complex data sets. To deal with this problem, some shape parameters have been added to Gumbel Type-II to improve the model flexibility.The current research study is motivated by producing new probability models by adding a shape and scale parameter. More surprisingly, there was a gap to modify the Gumbel type-II distribution by using the Marshall–Olkin family of distributions. In this study, we derive a new probability distribution by employing the cdf and pdf of Gumbel type-II in the Marshall–Olkin family of distributions called Marshall–Olkin Extended Gumbel Type-II (MOEGT-II) distribution.This article is organized as follows. In section2, we present Marshall–Olkin Extended Gumbel Type-II (MOEGT-II) distribution. In section 3, reliability analysis of MOEGT-II distribution is derived. Several mathematical properties of MOEGT-II distribution are derived in section 4. Measure of uncertainty and inequality of our proposed distribution are derived in section 5. Order statistics of the proposed distribution is derived in section 6. Estimation of the parameters by maximum likelihood is given in section 7, and in section 8, a simulation study is discussed. In section 9, we analyze three real data sets and compared our proposed distribution with different competent distributions. Finally, we conclude the paper in section 10.


## 2. Marshall–Olkin Extended Gumbel Type-II Distribution

The cdf of MOEGT-II distribution can be obtained using (3) and (4) in (1) as(6)GXx=1−1−exp−λx−ρθ1−1−α1−exp−λx−ρθ,x>0,α,λ,ρ,θ>0,where “λ” is the scale parameter and “α,” “θ,” and “ρ” are shape parameters.The corresponding pdf can be obtained by substituting (4) and (5) in (2), and we have(7)gXx=αθρλx−ρ−1exp−λx−ρ1−exp−λx−ρθ−11−1−α1−exp−λx−ρθ2,x>0,α,λ,ρ,θ>0.Figures1 and 2 define the plot of pdf with different values of parameters. We observed that the shape of density function is positively skewed, modified bathtub shape, and reverse J-shaped curve. From Figure 2, it can be observed that the peak of the curves shows a gradual rising behavior; however, a reverse bathtub curve can be also examined.Figure 1
Pdf for different values of parameterα, θ, and ρ with λ=1 for (a) ρ=1.2; (b) θ = 1.5; (c) α = 3.5.
(a)(b)(c)Figure 2
Cdf for different values of parameterα, θ, and ρ with λ=1 for (a) α = 4.5; (b) θ = 1.5; (c) ρ=0.5.
(a)(b)(c)


## 3. Reliability Analysis of Marshall–Olkin Extended Gumbel Type-II Distribution

In this section, the survival function (SF), hazard rate function (HRF), reverse hazard rate function (RHRF), and cumulative hazard rate function (CHRF) of MOEGT-II distribution are derived.The survival function of MOEGT-II distribution is(8)G¯x=α1−exp−λx−ρθ1−1−α1−exp−λx−ρθ,x>0,α,λ,ρ,θ>0,The graph of survival function of MOEGT-II distribution is shown in Figure3. It is demonstrated from Figures 3(a)–3(c) that curves are in a downward direction; however, varying the parameters produces a decreasing trend also.Figure 3
SF for different values of parameterα, θ, and ρ with λ=1 for (a) α = 2; (b) θ = 1.5; (c) ρ=1.5.
(a)(b)(c)The hazard rate function of MOEGT-II distribution is(9)hXx=θρλx−ρ−1exp−λx−ρ1−exp−λx−ρ−11−1−α1−exp−λx−ρθ,x>0,α,λ,ρ,θ>0.The graph of hazard rate function of MOEGT-II distribution is presented in Figure4. From Figure 4, it can be observed that for different values of parameters, the failure rate of MOEGT-II distribution is increasing, decreasing, reverse J-shaped, and upside down bathtub shape.Figure 4
HRF for different values of parameterα, θ, and ρ with λ=1 for (a) ρ=2.2; (b) θ = 1.5; (c) α = 2.0.
(a)(b)(c)The reverse hazard rate function of MOEGT-II distribution is(10)RXx=αθρλx−ρ−1exp−λx−ρ1−exp−λx−ρθ−11−1−exp−λx−ρθ−11−1−α1−exp−λx−ρθ.Figure 5
RHRF for different values of parameterα, θ, and ρ with λ=1 for (a) α = 0.01; (b) θ = 1.2; (c) ρ = 2.5.
(a)(b)(c)The graph of reverse hazard rate function of MOEGT-II distribution is presented in fig. Figures5(a)–5(c) clearly show that the reverse hazard rate function has a decreasing behavior.The cumulative hazard rate function of MOEGT-II distribution is as follows:(11)HXx=−lnα1−exp−λxρθ1−1−α1−exp−λx−ρθ.The graph of cumulative hazard rate function of MOEGT-II distribution is presented in Figure6. Figures 6(a)–6(c) clearly show that the cumulative hazard rate function has an increasing behavior.Figure 6
CHRF for different values of parameterα, θ, and ρ with λ=1 for (a) α = 15; (b) θ = 8.5; (c) ρ = 1.5.
(a)(b)(c)


## 4. Properties of MOEGT-II Distribution

### 4.1. Alternate Expression for Distribution Function

For any positive real number “a” and for z<1, we have the generalized binomial expansion:(12)1−z−a=∑k=0∞Γa+kΓazkk!.Consider the power series expansion:(13)1−zθ−1=∑k=0∞−1kΓθΓθ−kzkk!,which is valid for z<1 and any real noninteger θ.Applying the generalized binomial expansion (12) and power series expansion (13) twice in (6), we get(14)GXx=∑k=0∞wkexp−kλx−ρ,where(15)wk=∑i=0∞∑j=0∞1−αi−1j+ki!j!k!Γ1+iΓ1Γ2Γ2−jΓθi+j+1Γθi+j+1−k.


### 4.2. Alternate Expression for Density Function

Applying the generalized binomial expansion (12) and power series expansion (13) in equation (7), we get(16)gXx=αθρλx−ρ−1∑f=0∞tfexp−f+1λx−ρ,where(17)tf=∑i=0∞α¯i−1fi!f!Γ2+iΓ2Γθi+1Γθi+1−f.


### 4.3. Special Cases

For different values of parameters, the following distributions are obtained as special cases of MOEGT-II distribution.Case 1.
If substituteα=1 in (6), we get the Exponentiated Gumbel Type-II (EGT-II) distribution.(18)GXx=1−1−exp−λx−ρθ,x>0,λ,ρ,θ>0.Case 2.
If substituteθ=1 in (6), we get the Marshall–Olkin Gumbel Type-II (MOGT-II) distribution.(19)GXx=exp−λx−ρ1−1−α1−exp−λx−ρ,x>0,λ,α,ρ>0.Case 3.
If substituteα=θ=1 in (6), we get the Gumbel Type-II (GT-II) distribution.(20)GXx=exp−λx−ρ,x>0,λ,ρ>0.Case 4.
If substituteα=1 and λ=λρ in (6), we get the Exponentiated Frechet (EF) distribution.(21)GXx=1−1−exp−λ/xρθ,x>0,λ,θ,ρ>0.Case 5.
If substituteα=θ=1 and λ=λρ, we get the Frechet (F) distribution.(22)GXx=exp−λ/xρ,x>0,λ,ρ>0.Case 6.
If substituteθ=1 and λ=λρ, we get the Marshall–Olkin–Frechet (MOF) distribution.(23)GXx=exp−λ/xρ1−1−α1−exp−λ/xρ,x>0,λ,α,ρ>0.


### 4.4. Quantile Function

Quantile function gives the value of the random variable at a particular probability level in the given probability distribution. Quantile function is also known as inverse cumulative distribution.(24)xq=−1λln1−1−q1−α1−q−1/θ−1/p.The median for MOEGT-II distribution can be obtained by substitutingq=0.5 in the above expression:(25)median=−1λln1−1+α−1/θ−1/p,


### 4.5. Characteristic Function

The characteristic function of MOEGT-II distribution is obtained by substituting (16) in characteristic function formula:(26)ρxt=∫0∞expitxgXxdx=αθρλ∑f=0∞tf∫0∞expitxx−ρ−1exp−f+1λx−ρdx.Making the substitutionλx−ρ=u and after simplification, we get(27)ρxt=αθ∑f=0∞tf∫0∞expitλ1/ρu−1/ρexp−f+1udu.We have(28)exptx=∑j=0∞txjj!.Using the above expression, we get(29)∫0∞exp−attbdt=Γb+1ab+1.Using expressions (28)and (29) in the above equation, we get the characteristic function of MOEGT-II distribution.(30)ρxt=αθ∑f=0∞tf∑s=0∞itλ1/ρss!Γ−s/ρ+1f+1−s/ρ+1,under condition (s/ρ<1).


### 4.6. Moment Generating Function

Moment generating function can be defined as follows:(31)MXt=∑r=0∞trr!EXr,where(32)EXr=αθρλ∑f=0∞tf∫0∞xr−ρ+1exp−f+1λx−ρdx.Making the substitutionλx−ρ=u and using expansion (29) in the above equation, we get(33)EXr=αθλr/ρ∑f=0∞tfΓ1−r/ρf+11−r/ρ,under condition (r/ρ<1).Substituting (33) in (31), we have(34)MXt=∑r=0∞trr!αθλr/ρ∑f=0∞tfΓ1−r/ρf+11−r/ρ,which is the moment generating function of MOEGT-II distribution. For different values of “r,” i.e., r = 1, 2, 3, 4 in (33), we can obtain first four arbitrary moments:(35)μ1′=EX=αθλ1/ρ∑f=0∞tfΓ1−1/ρf+11−1/ρ,under the condition1ρ<1,μ2′=EX2=αθλ2/ρ∑f=0∞tfΓ1−2/ρf+11−2/ρ,under the condition2ρ<1,μ3′=EX3=αθλ3/ρ∑f=0∞tfΓ1−3/ρf+11−3/ρ,under the condition3ρ<1,μ4′=EX4=αθλ4/ρ∑f=0∞tfΓ1−4/ρf+11−4/ρ,under the condition4ρ<1.


### 4.7. Moments about Mean

Moments about means are(36)μ1=0,μ2=λ2/ραθ∑f=0∞tfΓ1−2/ρf+11−2/ρ−αθ∑f=0∞tfΓ1−1/ρf+11−1/ρ2, under the condition (r/ρ<1) for r = 1, 2.(37).μ3=λ3/ραθ∑f=0∞tfΓ1−3/ρf+11−3/ρ−3αθ∑f=0∞tfΓ1−2/ρf+11−2/ραθ∑f=0∞tfΓ1−1/ρf+11−1/ρ+2αθ∑f=0∞tfΓ1−1/ρf+11−1/ρ3,  under the condition (r/ρ<1) for r = 1, 2, 3.(38)μ4=λ4/ραθ∑f=0∞tfΓ1−4/ρf+11−4/ρ−4αθ∑f=0∞tfΓ1−3/ρf+11−3/ραθ∑f=0∞tfΓ1−1/ρf+11−1/ρ+6αθ∑f=0∞tfΓ1−2/ρf+11−2/ραθ∑f=0∞tfΓ1−1/ρf+11−1/ρ2−3αθ∑f=0∞tfΓ1−1/ρf+11−1/ρ4,  under the condition (r/ρ<1) for r = 1, 2, 3, 4.


### 4.8. Mean, Variance, Skewness, and Kurtosis

The mean and variance are provided in equations (35) and (36), and skewness and kurtosis of MOEGT-II distribution can be obtained as follows:(39)skewness=β1=μ3μ23/2,Using (36) and (37), we have(40)skewness=∑f=0∞tfΓ1−3/ρ/f+11−3/ρ−3αθ∑f=0∞tfΓ1−2/ρ/f+11−2/ρ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ+2α2θ2∑f=0∞tfΓ1−1/ρ/f+11−1/ρ3αθ∑f=0∞tfΓ1−2/ρ/f+11−2/ρ−αθ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ23/2, under the condition (r/ρ<1) for r = 1, 2, 3.(41)kurtosis=β2=μ4μ22.Using (38) and (36), we have(42)kurtosis=∑f=0∞tfΓ1−4/ρ/f+11−4/ρ−4αθ∑f=0∞tfΓ1−3/ρ/f+11−3/ρ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ+6α2θ2∑f=0∞tfΓ1−2/ρ/f+11−2/ρ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ2−3α3θ3∑f=0∞tfΓ1−1/ρ/f+11−1/ρ4αθ∑f=0∞tfΓ1−2/ρ/f+11−2/ρ−αθ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ22,  under the condition (r/ρ<1) for r = 1, 2, 3, 4.


### 4.9. Mean Deviation

The mean deviation can be defined from mean or a median.


#### 4.9.1. Mean Deviation about Mean

The mean deviation about mean is defined by Nadarajah and Kotz [29] with the following form:(43)M.Dx¯=2μFμ−∫0μxgxdx.Substitute (16) in the above expression,making the substitution f+1λx−ρ=s, and after simplification, we get(44)∫0μxgxdx=αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λμ−ρ.From (14), substituting x=μ, we get(45)Gμ=∑k=0∞wkexp−kλμ−ρ.By substituting (44) and (45) in (43), we get(46)M.Dx¯=2μ∑k=0∞wkexp−kλμ−ρ−2αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λμ−ρ,   under the condition (1/ρ<1) where μ is defined in (35).


#### 4.9.2. Mean Deviation about Median

The mean deviation about median is defined by Nadarajah and Kotz [29] with the following form:(47)M⋅Dx˜=Ex+2MFM−M−2∫0Mxgxdx.Substitutingμ=M in (44), we get(48)∫0Mxgxdx=αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λM−ρ.Using (35), (25) and (48), then (47) equation becomes(49)M.Dx˜=αθλ1/ρ∑f=0∞tfΓ1−1/ρf+11−1/ρ+2M−1λln1−1+α1/θ−1/ρ−M−2αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λM−ρ,under the condition (1/ρ<1) where M is median defined in (25).


### 4.10. Geometric Mean

The geometric mean of a random variable is defined by Abd El-Monsef and Ghoneim [30] with the following form:(50)logG⋅M=Elogx,where(51)Elogx=∫0∞logxgxdx.Substituting (16) in the above equation, making the substitution f+1λx−ρ=t, and after simplification, we get(52)logG⋅M=Elogx=αθ∑f=0∞tff+1−11ρlogf+1+1ρlogλ−1ρ∫0∞logte−tdt,as(53)∫0∞logte−tdt=0.5722.Substituting the result in the above equation, we get the geometric mean of MOEGT-II distribution:(54)logG⋅M=αθ∑f=0∞tff+1−11ρlogf+1+1ρlogλ−1ρ0.5722.


### 4.11. Harmonic Mean

The Harmonic mean is defined as(55)1H=∫0∞1xgxdx.Substituting (16) in the above expression, making the substitution f+1λx−ρ=t, and after simplification, we get(56)1H=αθλ−1/ρ∑f=0∞tff+1−1+1/ρ∫0∞t1+1/ρ−1exp−tdt.The Gamma function is(57)Γz=∫0∞xz−1exp−xdx.So, using Gamma function (57) in the above equation, we get the Harmonic mean of MOEGT-II distribution:(58)1H=αθλ−1/ρ∑f=0∞tff+1−1+1/ρΓ1+1ρ.


## 4.1. Alternate Expression for Distribution Function

For any positive real number “a” and for z<1, we have the generalized binomial expansion:(12)1−z−a=∑k=0∞Γa+kΓazkk!.Consider the power series expansion:(13)1−zθ−1=∑k=0∞−1kΓθΓθ−kzkk!,which is valid for z<1 and any real noninteger θ.Applying the generalized binomial expansion (12) and power series expansion (13) twice in (6), we get(14)GXx=∑k=0∞wkexp−kλx−ρ,where(15)wk=∑i=0∞∑j=0∞1−αi−1j+ki!j!k!Γ1+iΓ1Γ2Γ2−jΓθi+j+1Γθi+j+1−k.


## 4.2. Alternate Expression for Density Function

Applying the generalized binomial expansion (12) and power series expansion (13) in equation (7), we get(16)gXx=αθρλx−ρ−1∑f=0∞tfexp−f+1λx−ρ,where(17)tf=∑i=0∞α¯i−1fi!f!Γ2+iΓ2Γθi+1Γθi+1−f.


## 4.3. Special Cases

For different values of parameters, the following distributions are obtained as special cases of MOEGT-II distribution.Case 1.
If substituteα=1 in (6), we get the Exponentiated Gumbel Type-II (EGT-II) distribution.(18)GXx=1−1−exp−λx−ρθ,x>0,λ,ρ,θ>0.Case 2.
If substituteθ=1 in (6), we get the Marshall–Olkin Gumbel Type-II (MOGT-II) distribution.(19)GXx=exp−λx−ρ1−1−α1−exp−λx−ρ,x>0,λ,α,ρ>0.Case 3.
If substituteα=θ=1 in (6), we get the Gumbel Type-II (GT-II) distribution.(20)GXx=exp−λx−ρ,x>0,λ,ρ>0.Case 4.
If substituteα=1 and λ=λρ in (6), we get the Exponentiated Frechet (EF) distribution.(21)GXx=1−1−exp−λ/xρθ,x>0,λ,θ,ρ>0.Case 5.
If substituteα=θ=1 and λ=λρ, we get the Frechet (F) distribution.(22)GXx=exp−λ/xρ,x>0,λ,ρ>0.Case 6.
If substituteθ=1 and λ=λρ, we get the Marshall–Olkin–Frechet (MOF) distribution.(23)GXx=exp−λ/xρ1−1−α1−exp−λ/xρ,x>0,λ,α,ρ>0.


## 4.4. Quantile Function

Quantile function gives the value of the random variable at a particular probability level in the given probability distribution. Quantile function is also known as inverse cumulative distribution.(24)xq=−1λln1−1−q1−α1−q−1/θ−1/p.The median for MOEGT-II distribution can be obtained by substitutingq=0.5 in the above expression:(25)median=−1λln1−1+α−1/θ−1/p,


## 4.5. Characteristic Function

The characteristic function of MOEGT-II distribution is obtained by substituting (16) in characteristic function formula:(26)ρxt=∫0∞expitxgXxdx=αθρλ∑f=0∞tf∫0∞expitxx−ρ−1exp−f+1λx−ρdx.Making the substitutionλx−ρ=u and after simplification, we get(27)ρxt=αθ∑f=0∞tf∫0∞expitλ1/ρu−1/ρexp−f+1udu.We have(28)exptx=∑j=0∞txjj!.Using the above expression, we get(29)∫0∞exp−attbdt=Γb+1ab+1.Using expressions (28)and (29) in the above equation, we get the characteristic function of MOEGT-II distribution.(30)ρxt=αθ∑f=0∞tf∑s=0∞itλ1/ρss!Γ−s/ρ+1f+1−s/ρ+1,under condition (s/ρ<1).


## 4.6. Moment Generating Function

Moment generating function can be defined as follows:(31)MXt=∑r=0∞trr!EXr,where(32)EXr=αθρλ∑f=0∞tf∫0∞xr−ρ+1exp−f+1λx−ρdx.Making the substitutionλx−ρ=u and using expansion (29) in the above equation, we get(33)EXr=αθλr/ρ∑f=0∞tfΓ1−r/ρf+11−r/ρ,under condition (r/ρ<1).Substituting (33) in (31), we have(34)MXt=∑r=0∞trr!αθλr/ρ∑f=0∞tfΓ1−r/ρf+11−r/ρ,which is the moment generating function of MOEGT-II distribution. For different values of “r,” i.e., r = 1, 2, 3, 4 in (33), we can obtain first four arbitrary moments:(35)μ1′=EX=αθλ1/ρ∑f=0∞tfΓ1−1/ρf+11−1/ρ,under the condition1ρ<1,μ2′=EX2=αθλ2/ρ∑f=0∞tfΓ1−2/ρf+11−2/ρ,under the condition2ρ<1,μ3′=EX3=αθλ3/ρ∑f=0∞tfΓ1−3/ρf+11−3/ρ,under the condition3ρ<1,μ4′=EX4=αθλ4/ρ∑f=0∞tfΓ1−4/ρf+11−4/ρ,under the condition4ρ<1.


## 4.7. Moments about Mean

Moments about means are(36)μ1=0,μ2=λ2/ραθ∑f=0∞tfΓ1−2/ρf+11−2/ρ−αθ∑f=0∞tfΓ1−1/ρf+11−1/ρ2, under the condition (r/ρ<1) for r = 1, 2.(37).μ3=λ3/ραθ∑f=0∞tfΓ1−3/ρf+11−3/ρ−3αθ∑f=0∞tfΓ1−2/ρf+11−2/ραθ∑f=0∞tfΓ1−1/ρf+11−1/ρ+2αθ∑f=0∞tfΓ1−1/ρf+11−1/ρ3,  under the condition (r/ρ<1) for r = 1, 2, 3.(38)μ4=λ4/ραθ∑f=0∞tfΓ1−4/ρf+11−4/ρ−4αθ∑f=0∞tfΓ1−3/ρf+11−3/ραθ∑f=0∞tfΓ1−1/ρf+11−1/ρ+6αθ∑f=0∞tfΓ1−2/ρf+11−2/ραθ∑f=0∞tfΓ1−1/ρf+11−1/ρ2−3αθ∑f=0∞tfΓ1−1/ρf+11−1/ρ4,  under the condition (r/ρ<1) for r = 1, 2, 3, 4.


## 4.8. Mean, Variance, Skewness, and Kurtosis

The mean and variance are provided in equations (35) and (36), and skewness and kurtosis of MOEGT-II distribution can be obtained as follows:(39)skewness=β1=μ3μ23/2,Using (36) and (37), we have(40)skewness=∑f=0∞tfΓ1−3/ρ/f+11−3/ρ−3αθ∑f=0∞tfΓ1−2/ρ/f+11−2/ρ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ+2α2θ2∑f=0∞tfΓ1−1/ρ/f+11−1/ρ3αθ∑f=0∞tfΓ1−2/ρ/f+11−2/ρ−αθ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ23/2, under the condition (r/ρ<1) for r = 1, 2, 3.(41)kurtosis=β2=μ4μ22.Using (38) and (36), we have(42)kurtosis=∑f=0∞tfΓ1−4/ρ/f+11−4/ρ−4αθ∑f=0∞tfΓ1−3/ρ/f+11−3/ρ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ+6α2θ2∑f=0∞tfΓ1−2/ρ/f+11−2/ρ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ2−3α3θ3∑f=0∞tfΓ1−1/ρ/f+11−1/ρ4αθ∑f=0∞tfΓ1−2/ρ/f+11−2/ρ−αθ∑f=0∞tfΓ1−1/ρ/f+11−1/ρ22,  under the condition (r/ρ<1) for r = 1, 2, 3, 4.


## 4.9. Mean Deviation

The mean deviation can be defined from mean or a median.


### 4.9.1. Mean Deviation about Mean

The mean deviation about mean is defined by Nadarajah and Kotz [29] with the following form:(43)M.Dx¯=2μFμ−∫0μxgxdx.Substitute (16) in the above expression,making the substitution f+1λx−ρ=s, and after simplification, we get(44)∫0μxgxdx=αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λμ−ρ.From (14), substituting x=μ, we get(45)Gμ=∑k=0∞wkexp−kλμ−ρ.By substituting (44) and (45) in (43), we get(46)M.Dx¯=2μ∑k=0∞wkexp−kλμ−ρ−2αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λμ−ρ,   under the condition (1/ρ<1) where μ is defined in (35).


### 4.9.2. Mean Deviation about Median

The mean deviation about median is defined by Nadarajah and Kotz [29] with the following form:(47)M⋅Dx˜=Ex+2MFM−M−2∫0Mxgxdx.Substitutingμ=M in (44), we get(48)∫0Mxgxdx=αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λM−ρ.Using (35), (25) and (48), then (47) equation becomes(49)M.Dx˜=αθλ1/ρ∑f=0∞tfΓ1−1/ρf+11−1/ρ+2M−1λln1−1+α1/θ−1/ρ−M−2αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λM−ρ,under the condition (1/ρ<1) where M is median defined in (25).


## 4.9.1. Mean Deviation about Mean

The mean deviation about mean is defined by Nadarajah and Kotz [29] with the following form:(43)M.Dx¯=2μFμ−∫0μxgxdx.Substitute (16) in the above expression,making the substitution f+1λx−ρ=s, and after simplification, we get(44)∫0μxgxdx=αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λμ−ρ.From (14), substituting x=μ, we get(45)Gμ=∑k=0∞wkexp−kλμ−ρ.By substituting (44) and (45) in (43), we get(46)M.Dx¯=2μ∑k=0∞wkexp−kλμ−ρ−2αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λμ−ρ,   under the condition (1/ρ<1) where μ is defined in (35).


## 4.9.2. Mean Deviation about Median

The mean deviation about median is defined by Nadarajah and Kotz [29] with the following form:(47)M⋅Dx˜=Ex+2MFM−M−2∫0Mxgxdx.Substitutingμ=M in (44), we get(48)∫0Mxgxdx=αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λM−ρ.Using (35), (25) and (48), then (47) equation becomes(49)M.Dx˜=αθλ1/ρ∑f=0∞tfΓ1−1/ρf+11−1/ρ+2M−1λln1−1+α1/θ−1/ρ−M−2αθλ1/ρ∑f=0∞tff+11/ρ−1Γ1−1ρ,f+1λM−ρ,under the condition (1/ρ<1) where M is median defined in (25).


## 4.10. Geometric Mean

The geometric mean of a random variable is defined by Abd El-Monsef and Ghoneim [30] with the following form:(50)logG⋅M=Elogx,where(51)Elogx=∫0∞logxgxdx.Substituting (16) in the above equation, making the substitution f+1λx−ρ=t, and after simplification, we get(52)logG⋅M=Elogx=αθ∑f=0∞tff+1−11ρlogf+1+1ρlogλ−1ρ∫0∞logte−tdt,as(53)∫0∞logte−tdt=0.5722.Substituting the result in the above equation, we get the geometric mean of MOEGT-II distribution:(54)logG⋅M=αθ∑f=0∞tff+1−11ρlogf+1+1ρlogλ−1ρ0.5722.


## 4.11. Harmonic Mean

The Harmonic mean is defined as(55)1H=∫0∞1xgxdx.Substituting (16) in the above expression, making the substitution f+1λx−ρ=t, and after simplification, we get(56)1H=αθλ−1/ρ∑f=0∞tff+1−1+1/ρ∫0∞t1+1/ρ−1exp−tdt.The Gamma function is(57)Γz=∫0∞xz−1exp−xdx.So, using Gamma function (57) in the above equation, we get the Harmonic mean of MOEGT-II distribution:(58)1H=αθλ−1/ρ∑f=0∞tff+1−1+1/ρΓ1+1ρ.


## 5. Measure of Uncertainty and Inequality

In this section, the measure of uncertainty of MOEGT-II distribution has been derived including Renyi entropy andqth entropy. Lorenz and Bonferroni curves are also derived as a measure of inequalities of MOEGT-II distribution.


### 5.1. Renyi Entropy

Renyi entropy is defined as(59)IX;Rδ=11−δlogIXδ,where(60)IXδ=∫RgXδxdx,forδ>0andδ≠1.Substituting (7) in the above equation and then applying the generalized binomial expansion (12) and power series expansion (13), we get(61)IXδ=αθρλδ∑i=0∞∑j=0∞α¯i−1ji!j!Γ2δ+iΓ2δΓθδ+i−δ+1Γθδ+i−δ+1−j∫0∞x−ρ−1δexp−λx−ρδ+jdx.Making the substitutionλx−ρ=u and after simplification, we get(62)IXδ=αθδλ−δ/ρ+1/ρρδ−1∑i=0∞∑j=0∞α¯i−1ji!j!Γ2δ+iΓ2δΓθδ+i−δ+1Γθδ+i−δ+1−j∫0∞uδ+δ/ρ−1/ρ−1exp−δ+judu.As we know,(63)∫0∞exp−xθxα−1dx=Γαθα.Using expansion (63) in the above expression,(64)IXδ=αθδλ1/ρ1−δρ−1−δ∑i=0∞∑j=0∞α¯i−1ji!j!Γ2δ+iΓ2δΓθδ+i−δ+1Γθδ+i−δ+1−jΓ1/ρδ−1+δδ+j1/ρδ−1+δ.Applying log on the above equation and then substituting the resulting equation in (59), we get the Renyi entropy of MOEGT-II distribution:(65)IX;Rδ=δ1−δlogαθ+1ρlogλ−logρ+11−δlog∑i=0∞∑j=0∞α¯i−1ji!j!Γ2δ+iΓ2δΓθδ+i−δ+1Γθδ+i−δ+1−jΓ1/ρδ−1+δδ+j1/ρδ−1+δ,      under the condition (δ>1).It can be observed from Table1 that as values of parameters decrease, the value of Renyi entropy increases.Table 1
Renyi entropy of the MOEGT-II for different combinations of parameters.
αλρθδRenyi entropy6.54.93.34.21.80.0134.02.92.54.11.70.0983.52.82.03.21.50.3232.91.91.72.51.40.4262.81.41.52.31.30.4622.41.00.91.81.20.7522.30.80.71.61.150.8601.50.70.51.31.120.9271.40.50.41.01.100.9751.30.40.30.991.010.978


### 5.2.qth Entropy

q-entropy can be defined as(66)HXq=1q−1log1−IXq.Substitutingδ = q in equation (64) and then substituting the resulting equation in the above expression, we get the qth entropy of MOEGT-II distribution:(67)HXq=1q−1log1−αθqλ1/ρ1−qρ−1−q∑j=0∞tj,where(68)tj=∑i=0∞α¯i−1ji!j!Γ2q+iΓ2qΓθq+i−q+1Γθq+i−q+1−jΓ1/ρq−1+qq+j1/ρq−1+q.It can be observed from Table2 that the value of q-Renyi entropy decreases as the values of parameters decrease.Table 2
qth entropy of the MOEGT-II for different combinations of parameters.
αλρθδqth entropy2.00.73.13.00.916.8751.90.62.52.90.811.4241.50.52.32.80.710.0361.30.42.252.70.66.7221.20.372.222.60.596.4971.00.21.92.50.52.4720.90.11.42.30.441.5720.50.091.02.20.421.4730.40.050.92.10.401.3400.010.020.82.00.390.981


### 5.3. Lorenz Curve

L (x) denotes the Lorenz curve and is defined as(69)LX=G∗xEx,where(70)G∗x=∫0xwgwExdw.Substituting (16) and (69) in the above equation, we get(71)G∗x=ρλ1−1/ρ∑f=0∞tf∑f=0∞tfΓ1−1/ρ/f+11−1/ρ∫0xw−ρexp−f+1λw−ρdw.Making the substitutionf+1λw−ρ=z and after using upper incomplete gamma function, we get(72)G∗x=∑f=0∞tff+1−1+1/ρΓ1−1/ρ,f+1λx−ρ∑f=0∞tff+1−1+1/ρΓ1−1/ρ.Putting (72) and (35) in (34), we get the Lorenz curve of MOEGT-II distribution as(73)Lx=∑f=0∞tff+1−1+1/ρΓ1−1/ρ,f+1λx−ραθλ1/ρΓ1−1/ρ2∑f=0∞tff+1−1+1/ρ2.


### 5.4. Bonferroni Curve

B(x) denotes the Bonferroni Curve and is defined as(74)BX=LxGx.Substituting the result of Lorenz curve (73) and (14) in the above expression, we get the Bonferroni curve of MOEGT-II distribution as follows :(75)BX=∑f=0∞tff+1−1+1/ρΓ1−1/ρ,f+1λx−ραθλ1/ρΓ1−1/ρ2∑f=0∞tff+1−1+1/ρ2∑k=0∞wkexp−kλx−ρ.Figure7 defines the graph of Bonferroni and Lorenz curve.Figure 7
Graphs of Bonferroni and Lorenz curve.


## 5.1. Renyi Entropy

Renyi entropy is defined as(59)IX;Rδ=11−δlogIXδ,where(60)IXδ=∫RgXδxdx,forδ>0andδ≠1.Substituting (7) in the above equation and then applying the generalized binomial expansion (12) and power series expansion (13), we get(61)IXδ=αθρλδ∑i=0∞∑j=0∞α¯i−1ji!j!Γ2δ+iΓ2δΓθδ+i−δ+1Γθδ+i−δ+1−j∫0∞x−ρ−1δexp−λx−ρδ+jdx.Making the substitutionλx−ρ=u and after simplification, we get(62)IXδ=αθδλ−δ/ρ+1/ρρδ−1∑i=0∞∑j=0∞α¯i−1ji!j!Γ2δ+iΓ2δΓθδ+i−δ+1Γθδ+i−δ+1−j∫0∞uδ+δ/ρ−1/ρ−1exp−δ+judu.As we know,(63)∫0∞exp−xθxα−1dx=Γαθα.Using expansion (63) in the above expression,(64)IXδ=αθδλ1/ρ1−δρ−1−δ∑i=0∞∑j=0∞α¯i−1ji!j!Γ2δ+iΓ2δΓθδ+i−δ+1Γθδ+i−δ+1−jΓ1/ρδ−1+δδ+j1/ρδ−1+δ.Applying log on the above equation and then substituting the resulting equation in (59), we get the Renyi entropy of MOEGT-II distribution:(65)IX;Rδ=δ1−δlogαθ+1ρlogλ−logρ+11−δlog∑i=0∞∑j=0∞α¯i−1ji!j!Γ2δ+iΓ2δΓθδ+i−δ+1Γθδ+i−δ+1−jΓ1/ρδ−1+δδ+j1/ρδ−1+δ,      under the condition (δ>1).It can be observed from Table1 that as values of parameters decrease, the value of Renyi entropy increases.Table 1
Renyi entropy of the MOEGT-II for different combinations of parameters.
αλρθδRenyi entropy6.54.93.34.21.80.0134.02.92.54.11.70.0983.52.82.03.21.50.3232.91.91.72.51.40.4262.81.41.52.31.30.4622.41.00.91.81.20.7522.30.80.71.61.150.8601.50.70.51.31.120.9271.40.50.41.01.100.9751.30.40.30.991.010.978


## 5.2.qth Entropy

q-entropy can be defined as(66)HXq=1q−1log1−IXq.Substitutingδ = q in equation (64) and then substituting the resulting equation in the above expression, we get the qth entropy of MOEGT-II distribution:(67)HXq=1q−1log1−αθqλ1/ρ1−qρ−1−q∑j=0∞tj,where(68)tj=∑i=0∞α¯i−1ji!j!Γ2q+iΓ2qΓθq+i−q+1Γθq+i−q+1−jΓ1/ρq−1+qq+j1/ρq−1+q.It can be observed from Table2 that the value of q-Renyi entropy decreases as the values of parameters decrease.Table 2
qth entropy of the MOEGT-II for different combinations of parameters.
αλρθδqth entropy2.00.73.13.00.916.8751.90.62.52.90.811.4241.50.52.32.80.710.0361.30.42.252.70.66.7221.20.372.222.60.596.4971.00.21.92.50.52.4720.90.11.42.30.441.5720.50.091.02.20.421.4730.40.050.92.10.401.3400.010.020.82.00.390.981


## 5.3. Lorenz Curve

L (x) denotes the Lorenz curve and is defined as(69)LX=G∗xEx,where(70)G∗x=∫0xwgwExdw.Substituting (16) and (69) in the above equation, we get(71)G∗x=ρλ1−1/ρ∑f=0∞tf∑f=0∞tfΓ1−1/ρ/f+11−1/ρ∫0xw−ρexp−f+1λw−ρdw.Making the substitutionf+1λw−ρ=z and after using upper incomplete gamma function, we get(72)G∗x=∑f=0∞tff+1−1+1/ρΓ1−1/ρ,f+1λx−ρ∑f=0∞tff+1−1+1/ρΓ1−1/ρ.Putting (72) and (35) in (34), we get the Lorenz curve of MOEGT-II distribution as(73)Lx=∑f=0∞tff+1−1+1/ρΓ1−1/ρ,f+1λx−ραθλ1/ρΓ1−1/ρ2∑f=0∞tff+1−1+1/ρ2.


## 5.4. Bonferroni Curve

B(x) denotes the Bonferroni Curve and is defined as(74)BX=LxGx.Substituting the result of Lorenz curve (73) and (14) in the above expression, we get the Bonferroni curve of MOEGT-II distribution as follows :(75)BX=∑f=0∞tff+1−1+1/ρΓ1−1/ρ,f+1λx−ραθλ1/ρΓ1−1/ρ2∑f=0∞tff+1−1+1/ρ2∑k=0∞wkexp−kλx−ρ.Figure7 defines the graph of Bonferroni and Lorenz curve.Figure 7
Graphs of Bonferroni and Lorenz curve.


## 6. Order Statistics

In this section, the density ofkth order statistics for MOEGT-II distribution has been derived. The rth moments for the density of order statistics are obtained. The distribution of the median is also discussed.


### 6.1. Density ofkth Order Statistics

Letx1,x2,…,xn be a random sample of size n from MOEGT-II density function, g(x). Let Xk:n denote the kth order statistics. Then, the density function of kth order statistics of the random variable Xk:n can be obtained using(76)fXk:nx=n!k−1!n−k!fXxFXxk−11−FXxn−k.Substituting (6) and (7) in the above equation, after applying the generalized binomial expansion (12) and power series expansion (13), we get the final expression of density of kth order statistics of MOEGT-II distribution:(77)fXk:nx=∑h=0∞Ck:nhgx;λh+1,ρ,where(78)Ck:nh=n!k−1!n−k!αn+1−kθh+1∑i=0∞∑j=0∞α¯i−1j+hi!j!h!Γn+1+iΓn+1ΓkΓk−jΓθn−k+i+j+1Γθn−k+i+j+1−h,and(79)gx;λh+1,ρ=λh+1ρx−ρ−1exp−h+1λx−ρ.gx;λh+1,ρ denotes the Gumbel type-II density function with parameter λh+1 and ρ. So, the density function of the order statistics is simply an infinite linear combination of Gumbel type-II density.


### 6.2. Moments ofKth Order Statistics

Therth moment for the density of kth order statistics can be obtained as follows:(80)EXk:nr=∫0∞xrfK:nxdx.Substituting (77) in the above expression, making the substitution h+1λx−ρ=z, and using gamma function (57), we get the rth moment of the kth order statistic from MOEGT-II distribution as follows:(81)EXk:nr=λr/ρ∑h=0∞h+1r/ρCk:nhΓ1−rρ,under condition (r/ρ<1).For different values of “r,” i.e., r = 1, 2, 3, 4, in (81), we can obtain first four arbitrary moments of the kth order statistic.


### 6.3. Distribution of Median

In many situations, when data are skewed or when the outliers are present in the data, usually the median is found to be the best measure amongst all. The distribution of median can be obtained using the density ofkth order statistics for odd and even sample sizes.Whenn is odd, say n = 2l + 1, the median is the l+1th order statistic. By substituting k = l + 1 in the density function of kth order statistic, the distribution of median can be obtained.From (77), the density of kth order statistic and the density of median (m) are obtained as follows:(82)fl+1:nm=n!αnθλρm−ρ−1∑i=0∞∑j=0∞∑h=0∞α−lα¯i−1j+hi!j!h!l!n−l−1!Γn+1+iΓn+1Γl+1Γl+1−jΓθn−l+i+jΓθn−l+i+j−hexp−h+1λm−ρ.Whenn is even, say n=2l, the median is then obtained as xl+xl+1/2 and density can be obtained from the joint density function of lth and l+1th order statistics.The joint density function oflth and l+1th can be found using(83)fxl,xl+1=n!l−1!n−l−1!Fxll−1fxl1−Fxl+1n−l−1fxl+1.Substituting the distribution function (6) and density function (7) of MOEGT-II distribution in the above expression and after simplification, we get(84)fxl,xl+1=n!αn−l+1θ2ρ2λ2l−1!n−l−1!1−1−exp−λxl−ρθl−111−1−α1−exp−λxl−ρθl+1xl−ρ−1exp−λxl−ρxl+1−ρ−1exp−λxl+1−ρ1−exp−λxl−ρθ−11−exp−λxl+1−ρθn−l−111−1−α1−exp−λxl+1−ρθn−l+1,0<xl<xl+1<∞.Substituting(85)m=xl+xl+12⇒xl=2m−v,v=xl+1,we get(86)fm,v=2n!αn−l+1θ2ρ2λ2l−1!n−l−1!1−1−exp−λ2m−v−ρθl−111−1−α1−exp−λ2m−v−ρθl+12m−v−ρ−1exp−λ2m−v−ρv−ρ−1exp−λv−ρ1−exp−λ2m−v−ρθ−11−exp−λv−ρθn−l−111−1−α1−exp−λv−ρθn−l+1,0<m<v<∞.To get the marginal density of ‘m,’ we integrate the above expression w.r.t ‘v’; then, applying generalized binomial expansion (12) twice and power series expansion (13) thrice, we get(87)fm=2n!αn−l+1θ2ρ2λ2l−1!n−l−1!∑i=0∞∑j=0∞∑k=0∞∑p=0∞∑q=0∞α¯i+j−1k+p+qi!j!k!p!q!Γn−l+1+iΓn−l+1Γl+1+jΓl+1ΓlΓl−kΓθj+k+1Γθj+k+1−pΓθn+i−lΓθn+i−l−q∫m∞2m−v−ρ−1v−ρ−1exp−λp+12m−v−ρ+q+1v−ρdv.Now, using the expansion,(88)exp−x=∑i=0∞−1ixii!,(89)q+pn=∑j=0nnjpjqn−j.Substituting (88) and (89) in the above equation, then applying the binomial expansion (89) twice, then solving integration term, and after simplification, we get the distribution of median for MOEGT-II distribution:(90)fm=2−ρn!αn−l+1θ2ρ2λ2l−1!n−l−1!∑i=0∞∑j=0∞∑k=0∞∑p=0∞∑q=0∞∑r=0∞∑s=0r∑u=0ρs−r−1−12ρs−r−uλrα¯i+jq+1sp+1r−s−1k+p+q+r+ui!j!k!p!q!r!Γn−l+1+iΓn−l+1Γl+1+jΓl+1ΓlΓl−kΓθ(j+k+1Γθj+k+1−pΓθn+i−lΓθn+i−l−qrsρs−r−1−1um−2ρ−ρr−1ρs+ρ−u, for m>0.


## 6.1. Density ofkth Order Statistics

Letx1,x2,…,xn be a random sample of size n from MOEGT-II density function, g(x). Let Xk:n denote the kth order statistics. Then, the density function of kth order statistics of the random variable Xk:n can be obtained using(76)fXk:nx=n!k−1!n−k!fXxFXxk−11−FXxn−k.Substituting (6) and (7) in the above equation, after applying the generalized binomial expansion (12) and power series expansion (13), we get the final expression of density of kth order statistics of MOEGT-II distribution:(77)fXk:nx=∑h=0∞Ck:nhgx;λh+1,ρ,where(78)Ck:nh=n!k−1!n−k!αn+1−kθh+1∑i=0∞∑j=0∞α¯i−1j+hi!j!h!Γn+1+iΓn+1ΓkΓk−jΓθn−k+i+j+1Γθn−k+i+j+1−h,and(79)gx;λh+1,ρ=λh+1ρx−ρ−1exp−h+1λx−ρ.gx;λh+1,ρ denotes the Gumbel type-II density function with parameter λh+1 and ρ. So, the density function of the order statistics is simply an infinite linear combination of Gumbel type-II density.


## 6.2. Moments ofKth Order Statistics

Therth moment for the density of kth order statistics can be obtained as follows:(80)EXk:nr=∫0∞xrfK:nxdx.Substituting (77) in the above expression, making the substitution h+1λx−ρ=z, and using gamma function (57), we get the rth moment of the kth order statistic from MOEGT-II distribution as follows:(81)EXk:nr=λr/ρ∑h=0∞h+1r/ρCk:nhΓ1−rρ,under condition (r/ρ<1).For different values of “r,” i.e., r = 1, 2, 3, 4, in (81), we can obtain first four arbitrary moments of the kth order statistic.


## 6.3. Distribution of Median

In many situations, when data are skewed or when the outliers are present in the data, usually the median is found to be the best measure amongst all. The distribution of median can be obtained using the density ofkth order statistics for odd and even sample sizes.Whenn is odd, say n = 2l + 1, the median is the l+1th order statistic. By substituting k = l + 1 in the density function of kth order statistic, the distribution of median can be obtained.From (77), the density of kth order statistic and the density of median (m) are obtained as follows:(82)fl+1:nm=n!αnθλρm−ρ−1∑i=0∞∑j=0∞∑h=0∞α−lα¯i−1j+hi!j!h!l!n−l−1!Γn+1+iΓn+1Γl+1Γl+1−jΓθn−l+i+jΓθn−l+i+j−hexp−h+1λm−ρ.Whenn is even, say n=2l, the median is then obtained as xl+xl+1/2 and density can be obtained from the joint density function of lth and l+1th order statistics.The joint density function oflth and l+1th can be found using(83)fxl,xl+1=n!l−1!n−l−1!Fxll−1fxl1−Fxl+1n−l−1fxl+1.Substituting the distribution function (6) and density function (7) of MOEGT-II distribution in the above expression and after simplification, we get(84)fxl,xl+1=n!αn−l+1θ2ρ2λ2l−1!n−l−1!1−1−exp−λxl−ρθl−111−1−α1−exp−λxl−ρθl+1xl−ρ−1exp−λxl−ρxl+1−ρ−1exp−λxl+1−ρ1−exp−λxl−ρθ−11−exp−λxl+1−ρθn−l−111−1−α1−exp−λxl+1−ρθn−l+1,0<xl<xl+1<∞.Substituting(85)m=xl+xl+12⇒xl=2m−v,v=xl+1,we get(86)fm,v=2n!αn−l+1θ2ρ2λ2l−1!n−l−1!1−1−exp−λ2m−v−ρθl−111−1−α1−exp−λ2m−v−ρθl+12m−v−ρ−1exp−λ2m−v−ρv−ρ−1exp−λv−ρ1−exp−λ2m−v−ρθ−11−exp−λv−ρθn−l−111−1−α1−exp−λv−ρθn−l+1,0<m<v<∞.To get the marginal density of ‘m,’ we integrate the above expression w.r.t ‘v’; then, applying generalized binomial expansion (12) twice and power series expansion (13) thrice, we get(87)fm=2n!αn−l+1θ2ρ2λ2l−1!n−l−1!∑i=0∞∑j=0∞∑k=0∞∑p=0∞∑q=0∞α¯i+j−1k+p+qi!j!k!p!q!Γn−l+1+iΓn−l+1Γl+1+jΓl+1ΓlΓl−kΓθj+k+1Γθj+k+1−pΓθn+i−lΓθn+i−l−q∫m∞2m−v−ρ−1v−ρ−1exp−λp+12m−v−ρ+q+1v−ρdv.Now, using the expansion,(88)exp−x=∑i=0∞−1ixii!,(89)q+pn=∑j=0nnjpjqn−j.Substituting (88) and (89) in the above equation, then applying the binomial expansion (89) twice, then solving integration term, and after simplification, we get the distribution of median for MOEGT-II distribution:(90)fm=2−ρn!αn−l+1θ2ρ2λ2l−1!n−l−1!∑i=0∞∑j=0∞∑k=0∞∑p=0∞∑q=0∞∑r=0∞∑s=0r∑u=0ρs−r−1−12ρs−r−uλrα¯i+jq+1sp+1r−s−1k+p+q+r+ui!j!k!p!q!r!Γn−l+1+iΓn−l+1Γl+1+jΓl+1ΓlΓl−kΓθ(j+k+1Γθj+k+1−pΓθn+i−lΓθn+i−l−qrsρs−r−1−1um−2ρ−ρr−1ρs+ρ−u, for m>0.


## 7. Maximum Likelihood Estimation

Letx1,x2,…,xn be a random sample of size “n” from the MOEGT-II distribution defined in (7). Then, the likelihood function can be expressed as follows:(91)L=∏i=1nαθρλx−ρ−1exp−λx−ρ1−exp−λx−ρθ−11−1−α1−exp−λx−ρθ2.Takingln on both sides, we get(92)lnL=nlnα+nlnθ+nlnρ+nlnλ−ρ+1∑i=1nlnx−λ∑i=1nx−ρ+θ−1∑i=1nln1−exp−λx−ρ−2∑i=1nln1−1−α1−exp−λx−ρθ.Partially differentiating (92) w.r.t “α,” “θ,” “ρ,” and “λ,” we get(93)∂lnL∂α=nα−2∑i=1n1−exp−λx−ρθ1−1−α1−exp−λx−ρθ,∂lnL∂θ=nθ+∑i=1nln1−exp−λx−ρ+2∑i=1n1−α1−exp−λx−ρθln1−exp−λx−ρ1−1−α1−exp−λx−ρθ,∂lnL∂ρ=nρ−∑i=1nlnx+λ∑i=1nx−ρlnx−∑i=1nexp−λx−ρλx−ρlnxθ−11−exp−λx−ρ+2θ1−α1−exp−λx−ρθ−11−1−α1−exp−λx−ρθ,∂lnL∂λ=nλ−∑i=1nx−ρ+∑i=1nexp−λx−ρx−ρθ−11−exp−λx−ρ+2θ1−α1−exp−λx−ρθ−11−1−α1−exp−λx−ρθ.The maximum likelihood estimates of parametersα, θ, ρ, and λcan be obtained from the solution of equations given in (93). Since these are nonlinear equations and their solution in closed form cannot be obtained, therefore any numerical method can be used to obtain MLE.


## 8. Simulation Study

To check the flexibility of the MOEGT-II, we discuss the simulation study of the proposed distribution using Monte Carlo simulations. All results are obtained for 5000 Monte Carlo runs, and the simulations are performed using R. In each replication, random samples of size (n = 50, 100, 300, 500, and 1000) are generated for different combinations of parameters from MOEGT-II (α, θ, ρ, λ) distribution, and three different iterative procedures named as Nelder–Mead, Quasi-Newton, and Conjugate-Gradients (CG) method (also known as Fletcher–Reeves method) have been used. The means and standard deviations of the MLEs are listed in Tables 3 and 4.Table 3
MLE and their sample statistics for some simulated samples and true valuesα = 1, θ = 1, ρ = 1, and λ = 1.
Nelder–MeadQuasi-NewtonFletcher–ReevesNParametersMinMaxMeanStdMinMaxMeanStdMinMaxMeanStd50α0.08727.8661.3191.1490.0903.8841.0950.3870.1324.9521.0170.192θ0.2453.9291.0710.3860.1182.7221.0410.2610.2372.8691.0090.169ρ0.6491.7911.0160.1070.7721.7831.0240.0950.7841.4611.0250.087λ15.93442.2891.0071.2520.0262.3461.0050.1270.4753.8531.0150.125Iteration69279126244576883939989981343100α0.1188.5491.1410.5970.0555.0721.0360.2940.0582.2891.0110.119θ0.2482.6521.0340.2670.0992.5551.0190.1910.1191.7920.0090.114ρ0.7681.3551.0070.0740.8211.3391.0120.0640.8621.3341.0120.058λ7.72516.2901.0620.7050.4923.6871.0020.1170.4812.5981.0070.087Iteration6527911923196031125541195381846300α0.3662.9451.0390.2850.4472.3591.0170.1470.4061.5581.0030.065θ0.5131.6231.0080.1480.5851.4921.0070.0990.5661.3861.0040.067ρ0.8611.1391.0030.0420.8941.1541.0040.0360.9091.1191.0050.034λ5.7535.3491.0780.4210.5061.6081.0020.0560.5991.4281.0010.039Iteration612331122118502863744089982838500α0.4542.1291.0260.2230.5021.7591.0070.1160.5021.9051.0030.049θ0.6491.4851.0070.1170.6711.3881.0050.0810.7211.4011.0050.041ρ0.8981.1241.0020.0330.8990.1171.0030.0260.9121.3401.0020.019λ3.1333.7611.0910.3330.6521.6461.0040.0510.6611.6271.0030.029Iteration64234107222469112060402910835491000α0.5751.7471.0170.1520.5131.5091.0040.0810.6311.3021.0020.039θ0.7591.3311.0060.0820.6661.2691.0020.0550.7531.2151.0020.036ρ0.9191.0911.0010.0230.9341.0821.0020.0190.9361.0831.0010.018λ1.7853.7431.0930.2640.4241.4311.0010.0350.7381.2531.0010.026Iteration6322110520246061084545390183941Table 4
MLE and their sample statistics for some simulated samples and true valuesα = 2, θ = 4, ρ = 6, and λ = 5.
Nelder–MeadQuasi-NewtonFletcher–ReevesnParametersMinMaxMeanStdMinMaxMeanStdMinMaxMeanStd50α0.1541471.24.23439.6420.26811.6231.8130.6680.5047.3191.740.448θ0.45622.5415.1752.4480.93812.6684.2130.8182.0189.8914.2110.44ρ1.81534.8775.7731.5903.52511.0516.0990.623.7948.5826.2620.561λ3672.72083.43.16588.6781.71911.44.8650.6874.23511.5384.6640.664Iteration7950114138162954625492283541100α0.293430.191.9637.5770.1948.171.770.4790.5954.0141.7710.335θ0.52621.5144.9981.6141.33611.5594.2030.7132.0797.3124.1690.317ρ2.18632.6285.5940.8793.7028.4546.0260.4434.2918.1216.220.423λ854.336.55.20813.6511.4438.8154.8480.5941.4237.7644.7250.517Iteration7950113227199495039792184231300α0.5655.1461.6250.5050.51913.8511.810.3610.6593.6021.800.301θ2.4059.4434.9080.8962.18712.2884.1090.4112.6127.4024.1630.265ρ4.1877.4395.5010.4213.367.5956.0490.3354.5587.3286.2080.334λ6.86513.1485.5340.8770.15111.0464.8360.4260.8388.3224.7570.424Iteration7927312722213085158892485831500α0.733.8681.5960.3780.4114.3721.8050.3090.4944.3411.8920.261θ2.9697.924.8990.6961.98.2824.1060.3801.9457.9024.1050.236ρ4.4426.6895.4810.3254.1817.7186.0390.3233.9277.2406.0240.291λ3.7817.3435.5530.7151.5619.1434.820.3960.9286.9974.7240.387Iteration852551272520934526114990866301000α0.8932.7811.5590.2610.4734.3191.7790.2590.6614.0161.8240.256θ3.3716.9744.8610.502.0998.3834.0990.3762.5697.6904.1410.228ρ4.7216.5515.4760.2334.2567.1485.9840.2174.3337.3076.1820.215λ2.96413.4425.5660.6040.7598.3984.780.3900.3256.7054.7880.385Iteration83253128232134954641295086729From Tables3 and 4, it can be seen that with increasing sample size, the standard deviations for the estimates tend to decrease and the Fletcher–Reeves method provides a better estimate as compared with the Nelder–Mead method and Quasi-Newton method with increasing sample size.


## 9. Application

In this section, the MOEGT-II distribution is applied to three real data sets to illustrate the usefulness and applicability of the proposed model. The MLEs of the proposed model parameters are computed, and some goodness-of-fit statistics for the fitted models are computed inR by using the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method. The estimated cdfs of the data sets and the pdfs of the fitted distributions are plotted. In addition, the probability plot (P-P) is also discussed. We compare the MOEGT-II distribution with nine different models that are the Exponentiated Gumbel Type-II (EGT-II) distribution, Marshall–Olkin Gumbel Type-II (MOGT-II) distribution, Gumbel Type-II (GT-II) distribution, Marshall Olkin Frechet (MOF) distribution, Frechet (F) distribution, Burr III distribution, Log Logistic (LL) distribution, Beta Inverse Weibull (BIW) distribution, and Kumaraswamy Inverse Weibull (KIW) distribution.The first example is a data set from Nichols and Padgett [31] consisting of 100 observations on breaking stress of carbon fibres (in Gba). Recently, Afify et al. [32] and Saboor et al. [33] used these data to analyze the distributions. The data values are 3.7, 2.74, 2.73, 2.5, 3.6, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.9, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 4.2, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.7, 2.03, 1.8, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65, 2.341, 4.628, 1.244, 2.435, 4.806, 1.249, 2.464, 4.881, 1.262, 2.543, 5.140.The second example is a data set corresponding to service times for a particular windshield model including 63 observations that are classified as service times of windshields from Murthy et al. [34]. Recently, Tahir et al. [35] applied these data to the Weibull–Lomax distribution. The data values are 0.046, 1.436, 2.592, 0.140, 1.492, 2.600, 0.150, 1.580, 2.670, 0.248, 1.719, 2.717, 0.280, 1.794, 2.819, 0.313, 1.915, 2.820, 0.389, 1.920, 2.878, 0.487, 1.963, 2.950, 0.622, 1.978, 3.003, 0.900, 2.053, 3.102, 0.952, 2.065, 3.304, 0.996, 2.117, 3.483, 1.003, 2.137, 3.500, 1.010, 2.141, 3.622, 1.085, 2.163, 3.665, 1.092, 2.183, 3.695, 1.152, 2.240, 4.015, 1.183, 2.341, 4.628, 1.244, 2.435, 4.806, 1.249, 2.464, 4.881, 1.262, 2.543, 5.140.The third data set represents the fracture toughness of alumina (Al2O3) (in the units of MPa m1 = 2) by Nadarajah [36]. Recently, Arifa et al. [37] and Ghitany et al. [38] used these data to analyze the distributions. The data values are 5.5, 5, 4.9, 6.4, 5.1, 5.2, 5.2, 5, 4.7, 4, 4.5, 4.2, 4.1, 4.56, 5.01, 4.7, 3.13, 3.12, 2.68, 2.77, 2.7, 2.36, 4.38, 5.73, 4.35, 6.81, 1.91, 2.66, 2.61, 1.68, 2.04, 2.08, 2.13, 3.8, 3.73, 3.71, 3.28, 3.9, 4, 3.8, 4.1, 3.9, 4.05, 4, 3.95, 4, 4.5, 4.5, 4.2, 4.55, 4.65, 4.1, 4.25, 4.3, 4.5, 4.7, 5.15, 4.3, 4.5, 4.9, 5, 5.35, 5.15, 5.25, 5.8, 5.85, 5.9, 5.75, 6.25, 6.05, 5.9, 3.6, 4.1, 4.5, 5.3, 4.85, 5.3, 5.45, 5.1, 5.3, 5.2, 5.3, 5.25, 4.75, 4.5, 4.2, 4, 4.15, 4.25, 4.3, 3.75, 3.95, 3.51, 4.13, 5.4, 5, 2.1, 4.6, 3.2, 2.5, 4.1, 3.5, 3.2, 3.3, 4.6, 4.3, 4.3, 4.5, 5.5, 4.6, 4.9, 4.3, 3, 3.4, 3.7, 4.4, 4.9, 4.9, 5.Tables5–7 provide the MLEs and their corresponding standard errors (SEs) (in parentheses); and the result of AIC, CAIC, BIC, HQIC, log-likelihood, Cramer-von Mises W∗, Anderson–Darling A∗, and Kolmogorov–Smirnov (KS) statistics and its pvalue for each distribution are presented in Tables 8–10. The MOEGT-II distribution has the smallest values for the AIC, CAIC, BIC, HQIC, log-likelihood, Cramer-von Mises W∗, Anderson–Darling A∗, and Kolmogorov-Smirnov (KS) statistics as compared with other fitted distributions, suggesting that the MOEGT-II model provides the good fit. The estimated cdfs, histogram of three data sets, pdfs, and P-P are presented in Figures 8–13, respectively. It is clear from Tables 8–10 and Figures 8–13 that the MOEGT-II distribution provides better fit than EGT-II, MOGT-II, GT-II, MOF, Frechet (F), Burr III, LL, BIW, and KIW models for given three data sets.Table 5
MLEs and their SE (in parentheses) for breaking stress data.
DistributionαλρθMOEGT-II9.884 777 6 (13.054 577 0)5.089 054 8 (2.118 862 0)0.505 807 8 (0.181 352 4)54.932 699 8 (76.616 613 6)EGT-II—8.346 042 9 (1.091 707 0)0.507 850 7 (0.101 844 2)137.719 819 4 (149.254 856 0)MOGT-II158.282 337 3 (71.820 618 29)0.157 871 5 (0.057 381 76)3.635 606 7 (0.245 951 64)—GT-II—3.088 187 (0.327 115 2)1.769 024 (0.111 922 3)—MOF112.389 913 9 (57.785 004 16)0.627 062 2 (0.069 095 07)3.482 807 1 (0.248 425 82)—F—1.891 563 (0.113 794 3)1.769 022 (0.111 922 3)—Burr III——2.319 184 (0.149 555 1)5.221 175 (0.598 417 7)LL—43.357 123 (16.039 205 8)4.117 069 (0.343 907 9)—BIW28.855 093 4 (11.767 038 1)0.773 766 1 (0.133 844 5)0.377 652 0 (0.168 107 5)97.324 915 2 (74.619 980 4)KIW3.300 443 3 (2.257 001 01)0.476 214 9 (0.092 630 72)4.845 829 4 (1.876 991 23)176.087 174 1 (189.465 682 34)Table 6
MLEs and their SE (in parentheses) for service times of windshield data.
DistributionαλρθMOEGT-II17.343 521 7 (16.476 418 09)4.401 977 4 (1.267 586 86)0.208 865 7 (0.058 273 55)134.686 072 1 (148.932 304 57)EGT-II—5.877 250 7 (0.864 721 37)0.269 711 1 (0.042 099 34)118.691 077 6 (95.874 791 74)MOGT-II115.845 881 73 (59.952 866 916)0.020 763 57 (0.009 710 782)1.740 842 20 (0.146 055 081)—GT-II—0.943 541 8 (0.128 395 05)0.810 293 4 (0.065 628 02)—MOF207.295 920 4 (128.468 612 87)0.092 251 8 (0.023 996 99)1.829 382 9 (0.161 485 05)—F—0.930 597 0 (0.154 297 27)0.810 355 9 (0.065 629 43)—Burr III——1.564 679 (0.153 962 2)1.641 350 (0.208 898 8)LL—3.514 333 (0.933 951 1)2.135 245 (0.231 174 2)—BIW125.532 248 4 (58.098 105 90)0.458 660 7 (0.055 434 43)0.274 777 9 (0.096 170 21)105.112 800 2 (70.456 427 40)KIW2.709 499 7 (3.362 850 85)0.227 468 7 (0.031 733 62)5.441 804 4 (1.702 913 12)305.607 672 3 (234.649 978 71)Table 7
MLEs and their SE (in parentheses) for the fracture toughness of alumina data.
DistributionαλρθMOEGT-II15.294 400 5 (14.227 853 1)10.421 918 7 (2.633 296 1)0.789 969 1 (0.222 581 7)69.729 425 3 (70.055 174 4)EGT-II—19.184 216 8 (1.949 256 8)0.990 687 2 (0.114 450 2)67.827 826 2 (35.304 429 3)MOGT-II368.233 172 (298.072 646 0)61.918 359 (50.694 638 2)6.869 485 (0.511 673 3)—GT-II—48.477 859 (10.825 173 9)3.022 359 (0.184 929 5)—MOF0.004 179 529 (0.001 304 203)22.086 001 867 (3.392 175 442)1.033 172 128 (0.072 468 604)—F—3.611 630 (0.116 713 1)3.024 015 (0.185 130 9)—Burr III——3.058 00 (0.179 916)51.880 55 7 (11.180 150)LL—227.626 726 (54.064 342 1)3.933 087 (0.176 544 8)—BIW15.446 107 5 (3.741 849 2)1.509 306 4 (0.335 202 4)0.312 665 8 (0.150 726 5)91.427 467 5 (59.284 776 0)KIW11.252 737 6 (7.562 136 5)0.806 074 3 (0.109 693 1)2.683 321 0 (1.532 909 3)243.688 695 4 (189.885 147 0)Table 8
Several goodness-of-fit criteria for breaking stress data.
DistributionAICCAICBICHQIClW∗A∗KSp valueMOEGT-II290.86291.27301.27295.07−141.430.0750.4040.0660.772EGT-II293.57293.82301.38296.74−143.780.1560.8040.1020.253MOGT-II303.78304.04311.61306.95−148.890.2691.4320.1110.176GT-II350.28350.41355.49352.39−173.140.7564.3120.1770.004MOF305.56305.81313.37308.73−149.780.2731.4590.1060.22F350.28350.41355.49352.39−173.140.7564.3120.1770.003 6Burr III325.95326.07331.16328.055−160.970.5693.1730.1390.041LL296.56296.68301.77298.67−146.280.2391.2410.0910.388BIW335.42335.79346.26339.82−163.710.0880.5030.0820.444KIW335.96336.34346.79340.36−163.980.1080.5830.0830.427Table 9
Several goodness-of-fit criteria for service times of windshield data.
DistributionAICCAICBICHQICLW∗A∗KSp valueMOEGT-II209.33210.02217.91212.69−100.660.110.620.090.66EGT-II220.82221.23227.25223.35−107.410.321.910.160.06MOGT-II229.55229.96235.98232.08−111.770.432.520.150.11GT-II266.61266.81270.89268.29−131.310.995.420.220.003MOF227.26227.66233.69229.78−110.630.3992.3540.140.15F266.61266.81270.89268.29−131.310.995.420.220.004Burr III233.53233.73237.82235.22−114.760.533.120.190.02LL221.48221.78225.77223.17−108.740.331.980.120.31BIW220.33221.02228.91223.71−106.160.271.630.180.023KIW220.36221.05228.93223.73−106.180.281.710.160.07Table 10
Several goodness-of-fit criteria for the fracture toughness of alumina data.
DistributionAICCAICBICHQICLW∗A∗KSp valueMOEGT-II345.03345.38356.15349.54−168.520.070.520.050.92EGT-II359.03359.24367.36362.41−176.520.362.190.130.05MOGT-II363.19363.41371.54366.58−178.590.392.420.080.31GT-II425.79425.89431.35428.05−210.891.417.870.200.000 1MOF399.23399.44407.56402.62−196.620.945.490.140.03F425.79425.89431.35428.05−210.891.417.880.210.000 1Burr III423.54423.64429.09425.79−209.771.377.660.190.000 2LL409.49409.59415.05411.75−202.750.523.130.230.000 007BIW355.07355.42366.18359.58−173.540.271.640.130.051KIW356.12356.46367.23360.63−174.060.281.720.110.102Figure 8
(a) Graphs of estimated cdfs; (b) graphs of estimated pdfs for breaking stress data.
(a)(b)Figure 9
Graphs of P-P for breaking stress data.Figure 10
(a) Graphs of estimated cdfs; (b) graphs of estimated pdfs for service times of 63 aircraft windshield.
(a)(b)Figure 11
Graphs of P-P for service times of 63 aircraft windshield.Figure 12
(a) Graphs of estimated cdfs; (b) graphs of estimated pdfs for the fracture toughness of alumina data.
(a)(b)Figure 13
Graphs of P-P for the fracture toughness of alumina (Al2O3) data.


## 10. Conclusion

A new probability model called MOEGT-II is derived by employing the cdf of EGT-II in Marshall–Olkin family of distributions. The proposed model is important due to its modeling of various shapes of the hazard rate function such as increasing, decreasing, reverse J-shaped, and upside down bathtub. The special cases of MOEGT-II are called EGT-II, MOGT-II, GT-II, EF, F, and MOF distribution. Mathematical properties of the proposed distribution were studied including survival function, hazard rate function, reverse hazard rate function, Renyi entropy,qth entropy, Lorenz curve, and Bonferroni curve. Expressions for the kth order statistics and their rth moments are also derived. Performance of the maximum likelihood estimates is also investigated through Monte Carlo simulation. We observed that with the increase in the sample size, the better results for maximum likelihood estimates are obtained. Also, the Conjugate-Gradients (CG) method (Fletcher–Reeves method) provides better estimates as compared with the Nelder–Mead method and Quasi-Newton method with increasing sample size. At the end, we have considered three data sets, and it has been shown that MOEGT-II distribution provides better performance than EGT-II, MOGT-II, GT-II, MOF, F, Burr III, LL, BIW, and KIW distributions.



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