# The Half-Logistic Generalized Weibull Distribution

Authors: Masood Anwar; Amna Bibi
Journal: Journal of Probability and Statistics
 (2018)

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## Abstract

A new three-parameter generalized distribution, namely, half-logistic generalized Weibull (HLGW) distribution, is proposed. The proposed distribution exhibits increasing, decreasing, bathtub-shaped, unimodal, and decreasing-increasing-decreasing hazard rates. The distribution is a compound distribution of type I half-logistic-G and Dimitrakopoulou distribution. The new model includes half-logistic Weibull distribution, half-logistic exponential distribution, and half-logistic Nadarajah-Haghighi distribution as submodels. Some distributional properties of the new model are investigated which include the density function shapes and the failure rate function, raw moments, moment generating function, order statistics, L-moments, and quantile function. The parameters involved in the model are estimated using the method of maximum likelihood estimation. The asymptotic distribution of the estimators is also investigated via Fisher’s information matrix. The likelihood ratio (LR) test is used to compare the HLGW distribution with its submodels. Some applications of the proposed distribution using real data sets are included to examine the usefulness of the distribution.


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## Body

## 1. Introduction

Statistical distributions are the basic aspects of all parametric statistical techniques including inference, modeling, survival analysis, and reliability. For the analysis of lifetime data, it is an important task to fit the data by a statistical model. A number of lifetime distributions have been developed in literature for this purpose. The widely used lifetime models usually have a limited range of behaviors. Such type of distributions cannot give a better fit to model all the practical situations. Recently, several authors have developed a number of families of statistical models by applying different techniques. Various techniques have been introduced in the literature to derive new flexible models as discussed by Lai [1].Marshall and Olkin [2] introduced an effective technique to extend a family of distributions by addition of another parameter. By applying this technique, they generalized the exponential and the Weibull distributions. Al-Zahrani and Sagor [3] proposed the Poisson Lomax model by compounding the Poisson Lomax distributions. Bidram and Nadarajah [4] introduced the exponentiated EG2 distribution by using the method of resilience parameter. Kus [5] considered compounding of Poisson and exponential distribution. Gurvich et al. [6] generalized the Weibull distribution offering a new distribution function elucidating a wide range of functional forms of the effect of size on the strength distribution, using a simple method of evaluation of the basic statistical parameters. Nadarajah and Kotz [7], Lai et al. [8], Lai et al. [9], and Xie et al. [10] further discussed some modifications of the Weibull model.In this paper, another extension of the extended Weibull distribution is introduced using half-logistic-G generator. So we propose the half-logistic generalized Weibull (HLGW) distribution without adding any extra parameter to the baseline model. The new model is the compound distribution of two previously known distributions, one of which follows the class proposed by Gurvich et al. [6] and the other is type I half-logistic-G model. The proposed model is able to depict more complex hazard rates and provides a good alternate to the Weibull distribution that does not exhibit upside-down bathtub-shaped or unimodal failure rates.Dimitrakopoulou et al. [11] established a three-parameter lifetime model with PDF(1)gx;ω,η,γ=ωηγxη-11+γxηω-1exp⁡1-1+γxηω,where x>0 and ω,η>0 are the shape parameters and γ>0 is a scale parameter. The CDF corresponding to (1) is(2)Gx;ω,η,γ=1-exp⁡1-1+γxηω.In this paper, a three-parameter lifetime model is presented which is the compound model of the previously known models introduced by Dimitrakopoulou et al. [11] and half-logistic-G (HL-G) distribution called half-logistic generalized Weibull (HLGW) distribution. The half-logistic-G distribution is presented by Cordeiro et al. [12] with the CDF (3)Gx;γ,θ=∫0-ln⁡1-Fx;θ2γe-γx1+e-γx2dx=1-1-Fx;θγ1+1-Fx;θγ,where F(x;θ) is the CDF of the baseline distribution and γ>0 is the shape parameter. As a special case, for γ=1, the TIHL-G is the half-logistic-G (HL-G) model with cumulative distribution function(4)Gx;θ=Fx;θ1+F¯x;θ.The corresponding PDF to (4) is given by(5)gx;θ=2fx;θ1+F¯x;θ2,where f(x)=d/dxF(x) and F¯(x;θ)=1-F(x;θ).The rest of the paper is unfolded as follows. Section2 contains the introduction of the half-logistic generalized Weibull (HLGW) distribution and provides the plots of its density function. Section 3 explores the distributional properties of the HLGW model. In Section 4, the method of maximum likelihood estimation is used to obtain the estimators of unknown parameters. The asymptotic distribution of the estimators is also investigated in this section via Fisher’s information matrix. A simulation study is discussed in Section 5 to check out the accuracy of point and interval estimates of the HLGW parameters. Section 6 involves some applications of the HLGW distribution using lifetime data sets to examine the fitness of the proposed model. Section 7 provides concluding remarks about the paper.


## 2. The Half-Logistic Generalized Weibull Distribution

Substitution of (1) and (2) in (5) results the following PDF of the HLGW distribution:(6)gx;ω,η,γ=2ωηγxη-11+γxηω-1exp⁡1-1+γxηω1+exp⁡1-1+γxηω2,forx>0.The CDF associated with (6) is as follows:(7)Gx=1-exp⁡1-1+γxηω1+exp⁡1-1+γxηω,The parameters ω,η>0 are the shape parameters and γ>0 is a scale parameter. From now on, a random variable X with PDF (6) will be written as X~HLGW(ω,η,γ).


## 3. Distributional Properties

This section deals with the investigation of the distributional properties of HLGW distribution. The statistical properties include the plots of the density function, the failure rate function, raw moments, moment generating function, order statistics, L-moments, and quantile function.


### 3.1. Special Cases

The HLGW distribution includes the following distributions as special cases:(i)
Forω=1, the HLGW model reduces to the half-logistic Weibull (HLW) model with the PDF (8)gx;η,γ=2ηγxη-1exp⁡-γxη1+exp⁡-γxη2,whereη is the shape parameter and γ is the scale parameter. For -γ=α, the half-logistic Weibull model is also called the power half-logistic distribution (PHLD) proposed and studied by Krishnarani [13].(ii)
Forω=η=1, the HLGW model generates a new model, the half-logistic exponential (HLE) model, with scale parameter γ and the PDF (9)gx;γ=2γexp⁡-γx1+exp⁡-γx2.(iii)
Forη=1, the HLGW model gives another new model, the half-logistic Nadarajah-Haghighi (HLNH) model, with the PDF (10)gx;ω,γ=2ωγ1+γxω-1exp⁡1-1+γxω1+exp⁡1-1+γxω2,whereω and γ are the shape and scale parameters, respectively.


### 3.2. The Shapes of HLGW Distribution

The following are the conditions under which the PDF of the HLGW distribution (6) shows different behaviors:(i)
Forη<1, the PDF is monotone decreasing with limx→0+g(x;θ)=∞, limx→∞g(x;θ)=0.(ii)
Forη=1, the same shape is exhibited with limx→0+g(x;θ)=ωγ/2 and limx→∞g(x;θ)=0.(iii)
Forη>1, the PDF has the value zero at the origin; then it increases to a higher value and then decreases, falling towards the value of zero at infinity.Various behaviors of the PDF are shown in Figure1, for some parameter values; γ=1. The mode of density (6) can be obtained from d/dx[log⁡(g(x))]x=x0=0.Figure 1
The shapes of the PDF of the HLGW distribution.


### 3.3. Hazard Rate Function (HRF)

ForX is a random variable, the HRF is given as h(x)=g(x)/G¯(x), where G¯=1-G represents the survival function given by(11)G¯x=2exp⁡1-1+γxηω1+exp⁡1-1+γxηω.The HRF of X~HLGW(ω,η,γ) can be written as(12)hx;ω,η,γ=ωηγxη-11+γxηω-11+exp⁡1-1+γxηω,for x>0, demonstrating different shapes for different parameter values. By differentiating (12), it can be easily checked that (a)
forω=1 and η=1, the value of HRF h(x) is zero at the origin; then it increases to its maximum; after that it is constant,(b)
forω≥1 and η>1 or ω<1 and η≤1, h(x) has increasing or decreasing behavior,(c)
forω≥1 and η<1,
(i)
ifωη≤1, h(x) is decreasing or decreasing-increasing-decreasing (DID),(ii)
ifωη>1, h(x) is increasing or bathtub-shaped,(d)
forω≤1 and η>1,
(i)
ifωη<1, h(x) has upside-down bathtub shape (unimodal),(ii)
ifωη≥1, h(x) is monotone increasing.The different hazard shapes are shown in Figure2 for different parameter values. By restricting ω(n+1)-1∈N, the failure rate function (12) can also be depicted as(13)hx;ω,η,γ=∑k=1∞∑n=0∞∑r=0ωn+1-1-1n+k-1ek-1k-1nn+1!γrηrxηr-1,for x>0, where γr=ω(n+1)r+1γr+1 and ηr=η(r+1). Thus the HRF can be written as the sum of ω terms and hence (13) is the failure rate function of a series system of ω components.Figure 2
The HRF of HLGW distribution for different parameter values.


### 3.4. Moments

In this section, therth moment μr′=E[Xr] of the HLGW model is presented as an infinite sum representation. The first four moments for r=1,2,3,4 have been calculated accordingly.Theorem 1.
LetX~HLGW(ω,η,γ) be a random variable, where ω,η,γ>0, and the rth moment of X about the origin is as follows:(14)μr′=EXr=2ωγr/η∑j=0r/η∑k=1∞∑n=0∞rηjekkn+1-1j+k+n+r/ηj+ωn+1n!,where r=1,2,3,4 and r/η∈N.Corollary 2.
LetX~HLGW(ω,η,γ) be a random variable, where ω,η,γ>0 and r/η∈N. The first four moments of the random variable are(15)μ′=EX=2ωγ1/η∑j=01/η∑k=1∞∑n=0∞1ηjekkn+1-1j+k+n+1/ηj+ωn+1n!μ2′=EX2=2ωγ2/η∑j=02/η∑k=1∞∑n=0∞2ηjekkn+1-1j+k+n+2/ηj+ωn+1n!μ3′=EX3=2ωγ3/η∑j=03/η∑k=1∞∑n=0∞3ηjekkn+1-1j+k+n+3/ηj+ωn+1n!μ4′=EX4=2ωγ4/η∑j=04/η∑k=1∞∑n=0∞4ηjekkn+1-1j+k+n+4/ηj+ωn+1n!.


### 3.5. Moment Generating Function (MGF)

The MGF ofX is retrieved from MX(t)=E[etX]=∫etxg(x)dx.Theorem 3.
letX~HLGW(ω,η,γ) be a random variable, and the moment generating function (MGF) of X is given by(16)MXt=2ωγj/η∑k=1∞∑n=0∞∑j=0n∑i=0j/ηnjjηiekkn-j+1tj-1i+j+k+n+j/ηi+ωn-j+1n!,where j/η∈N.


### 3.6. Order Statistics

LetX1,X2,…,Xm be a random sample of size m from a distribution with PDF g(x) and CDF G(x) and X1:m,X2:m,…,Xm:m are the analogous order statistics. The PDF and CDF of Xr:m, 1⩽r⩽m, are(17)gr:mx=1Br,m-r+1gxGxr-11-Gxm-r=1Br,m-r+1gx∑i=0m-rmjm-ri-1iGxi+r-1,Gr:m=∑k=rmmkGxk1-Gxm-k,where B(r,m-r+1) is the beta function.Theorem 4.
Letg(x) and G(x) be the PDF and CDF of random variable X~HLGW(ω,η,γ); then the PDF of Xr:m is(18)gr:mx=2ωηγCr:m∑i=0m-rm-ri-1ixη-11+γxηω-1exp⁡1-1+γxηω1+exp⁡1-1+γxηω2×1-exp⁡1-1+γxηω1+exp⁡1-1+γxηωi+r-1,where Cr:m=Br,m-r+1-1.
The CDF corresponding to (18) is (19)Gr:mx=∑k=rmmk1-exp⁡1-1+γxηω1+exp⁡1-1+γxηωk2exp⁡1-1+γxηω1+exp⁡1-1+γxηωm-k.


### 3.7. L-Moments

Suppose that we have a random sampleX1,X2,…,Xn collected from X~HLGW(ω,η,γ). The rth population L-moments are as follows: (20)EXr:n=∫0∞xgXr:ndx=∫0∞x∑i=0n-rn-ri-1i2ωηγCr:nxη-11+γxηω-1e1-1+γxηω1+e1-1+γxηω2×1-e1-1+γxηω1+e1-1+γxηωi+r-1dx.We use the substitution y=1+γxη, so x=y-1/γ1/η and dx=1/γηy-1/γ1/η-1dy, where 1/η∈N. Therefore, we get (21)EXr:n=2ωCr:n∑i=0n-r∑k=0i+r-1∑j=0∞∑l=01/η∑m=0∞n-rii+r-1k-i+r+1j1ηl×-1γ1/η-1j+k+l+mej+k+1j+k+1mm!∫1∞yl+mω+ω-1dy.By working out the integration, we arrive at the following formula:(22)EXr:n=2ω∑i=0n-r∑k=0i+r-1∑j=0∞∑l=01/η∑m=0∞-1γ1/ηj+k+1mej+k+1l+ωm+1m!Aijkl,where (23)Aijkl=Cr:n-1j+k+l+m+1n-rii+r-1k-i+r+1j1ηl.The relation (22) can be used to find out the first L-moments of Xr:n; that is, for r=n=1 we get λ1=E[X1:1]. (24)λ1=2ω∑j=0∞∑m=0∞∑l=01/η-2j1ηl-1γ1/η-1j+l+m+1ej+1j+1ml+ωm+1m!.The other two moments, λ2 and λ3, are, respectively, given by (25)λ2=4ω∑j=0∞∑l=01/η∑m=0∞1ηl-1/γ1/ηl+ωm+1m!×∑i=01∑k=0i1iik-2-ij-1j+k+l+m+1ej+k+lj+k+lm+∑k=011k-3j-1j+k+l+m+1ej+k+1j+k+1m.λ3=2ω∑j=0∞∑l=01/η∑m=0∞1ηl-1/γ1/η-1j+k+l+m+1ej+k+1j+k+1ml+ωm+1m!×∑i=02∑k=0i2iik-2-ij+3!∑i=01∑k=0i+11ii+1k-3-ij+3∑k=022k-4-ij


### 3.8. Quantile Function

ForX to be a random variable with the PDF (6), the quantile function Q(u) is (26)Qu=inf⁡x∈R:Gx⩾u,where0<u<1.The above relation is used to find the quantile function of HLGW distribution. Therefore, we have(27)Qu=1γ1-ln⁡1-u1+u1/ω-11/η.Hence, the generator forX can be given by the following algorithm:(1)
GenerateU~uniform(0,1).(2)
Use (27) and obtain an outcome of X by X=Q(U).By using the quantile function (27), we can examine the Bowley skewness [14] and Moors kurtosis [15] for HLGW as follows: (28)sk=Q3/4+Q1/4-2Q2/4Q3/4-Q1/4κ=Q3/8-Q1/8+Q7/8-Q5/8Q3/4-Q1/4.Table 1 illustrates the values of skewness and kurtosis for the HLGW model for some values of ω, η, and γ. It can be noted that the skewness and kurtosis are free of parameter γ and they are decreasing functions of the parameters ω and η.Table 1
Skewness and kurtosis of HLGW for different values ofω, η, and γ.
γ
ω
η
=
0.5
η
=
1.0
η
=
2.0
s
k
k
u
s
k
k
u
s
k
k
u
0.5
0.5
0.6530
2.4642
0.3366
1.4540
0.1303
1.2492
1.0
0.4634
1.6286
0.1808
1.2395
0.0197
1.2074
1.5
0.3913
1.4463
0.1264
1.1952
−0.0196
1.2045
2.0
0.3541
1.3700
0.0989
1.1779
−0.0396
1.2054
2.5
0.3314
1.3286
0.0824
1.1691
−0.0517
1.2067
1.5
0.5
0.6530
2.4642
0.3366
1.4540
0.1303
1.2492
1.0
0.4634
1.6286
0.1808
1.2395
0.0197
1.2074
1.5
0.3913
1.4463
0.1264
1.1952
−0.0196
1.2045
2.0
0.3541
1.3700
0.0989
1.1779
−0.0396
1.2054
2.5
0.3314
1.3286
0.0824
1.1691
−0.0517
1.2067
2.0
0.5
0.6530
2.4642
0.3366
1.4540
0.1303
1.2492
1.0
0.4634
1.6286
0.1808
1.2395
0.0197
1.2074
1.5
0.3913
1.446
0.1264
1.1952
−0.0196
1.2045
2.0
0.3541
1.3700
0.0989
1.1779
−0.0396
1.2054
2.5
0.3314
1.3286
0.0824
1.1691
−0.0517
1.2067
3.0
0.5
0.6530
2.4642
0.3366
1.4540
0.1303
1.2492
1.0
0.4634
1.6286
0.1808
1.2395
0.0197
1.2074
1.5
0.3913
1.4463
0.1264
1.1956
−0.0196
1.2045
2.0
0.3541
1.3700
0.0989
1.1779
−0.0396
1.2054
2.5
0.3314
1.3286
0.0824
1.1691
−0.0517
1.2067


## 3.1. Special Cases

The HLGW distribution includes the following distributions as special cases:(i)
Forω=1, the HLGW model reduces to the half-logistic Weibull (HLW) model with the PDF (8)gx;η,γ=2ηγxη-1exp⁡-γxη1+exp⁡-γxη2,whereη is the shape parameter and γ is the scale parameter. For -γ=α, the half-logistic Weibull model is also called the power half-logistic distribution (PHLD) proposed and studied by Krishnarani [13].(ii)
Forω=η=1, the HLGW model generates a new model, the half-logistic exponential (HLE) model, with scale parameter γ and the PDF (9)gx;γ=2γexp⁡-γx1+exp⁡-γx2.(iii)
Forη=1, the HLGW model gives another new model, the half-logistic Nadarajah-Haghighi (HLNH) model, with the PDF (10)gx;ω,γ=2ωγ1+γxω-1exp⁡1-1+γxω1+exp⁡1-1+γxω2,whereω and γ are the shape and scale parameters, respectively.


## 3.2. The Shapes of HLGW Distribution

The following are the conditions under which the PDF of the HLGW distribution (6) shows different behaviors:(i)
Forη<1, the PDF is monotone decreasing with limx→0+g(x;θ)=∞, limx→∞g(x;θ)=0.(ii)
Forη=1, the same shape is exhibited with limx→0+g(x;θ)=ωγ/2 and limx→∞g(x;θ)=0.(iii)
Forη>1, the PDF has the value zero at the origin; then it increases to a higher value and then decreases, falling towards the value of zero at infinity.Various behaviors of the PDF are shown in Figure1, for some parameter values; γ=1. The mode of density (6) can be obtained from d/dx[log⁡(g(x))]x=x0=0.Figure 1
The shapes of the PDF of the HLGW distribution.


## 3.3. Hazard Rate Function (HRF)

ForX is a random variable, the HRF is given as h(x)=g(x)/G¯(x), where G¯=1-G represents the survival function given by(11)G¯x=2exp⁡1-1+γxηω1+exp⁡1-1+γxηω.The HRF of X~HLGW(ω,η,γ) can be written as(12)hx;ω,η,γ=ωηγxη-11+γxηω-11+exp⁡1-1+γxηω,for x>0, demonstrating different shapes for different parameter values. By differentiating (12), it can be easily checked that (a)
forω=1 and η=1, the value of HRF h(x) is zero at the origin; then it increases to its maximum; after that it is constant,(b)
forω≥1 and η>1 or ω<1 and η≤1, h(x) has increasing or decreasing behavior,(c)
forω≥1 and η<1,
(i)
ifωη≤1, h(x) is decreasing or decreasing-increasing-decreasing (DID),(ii)
ifωη>1, h(x) is increasing or bathtub-shaped,(d)
forω≤1 and η>1,
(i)
ifωη<1, h(x) has upside-down bathtub shape (unimodal),(ii)
ifωη≥1, h(x) is monotone increasing.The different hazard shapes are shown in Figure2 for different parameter values. By restricting ω(n+1)-1∈N, the failure rate function (12) can also be depicted as(13)hx;ω,η,γ=∑k=1∞∑n=0∞∑r=0ωn+1-1-1n+k-1ek-1k-1nn+1!γrηrxηr-1,for x>0, where γr=ω(n+1)r+1γr+1 and ηr=η(r+1). Thus the HRF can be written as the sum of ω terms and hence (13) is the failure rate function of a series system of ω components.Figure 2
The HRF of HLGW distribution for different parameter values.


## 3.4. Moments

In this section, therth moment μr′=E[Xr] of the HLGW model is presented as an infinite sum representation. The first four moments for r=1,2,3,4 have been calculated accordingly.Theorem 1.
LetX~HLGW(ω,η,γ) be a random variable, where ω,η,γ>0, and the rth moment of X about the origin is as follows:(14)μr′=EXr=2ωγr/η∑j=0r/η∑k=1∞∑n=0∞rηjekkn+1-1j+k+n+r/ηj+ωn+1n!,where r=1,2,3,4 and r/η∈N.Corollary 2.
LetX~HLGW(ω,η,γ) be a random variable, where ω,η,γ>0 and r/η∈N. The first four moments of the random variable are(15)μ′=EX=2ωγ1/η∑j=01/η∑k=1∞∑n=0∞1ηjekkn+1-1j+k+n+1/ηj+ωn+1n!μ2′=EX2=2ωγ2/η∑j=02/η∑k=1∞∑n=0∞2ηjekkn+1-1j+k+n+2/ηj+ωn+1n!μ3′=EX3=2ωγ3/η∑j=03/η∑k=1∞∑n=0∞3ηjekkn+1-1j+k+n+3/ηj+ωn+1n!μ4′=EX4=2ωγ4/η∑j=04/η∑k=1∞∑n=0∞4ηjekkn+1-1j+k+n+4/ηj+ωn+1n!.


## 3.5. Moment Generating Function (MGF)

The MGF ofX is retrieved from MX(t)=E[etX]=∫etxg(x)dx.Theorem 3.
letX~HLGW(ω,η,γ) be a random variable, and the moment generating function (MGF) of X is given by(16)MXt=2ωγj/η∑k=1∞∑n=0∞∑j=0n∑i=0j/ηnjjηiekkn-j+1tj-1i+j+k+n+j/ηi+ωn-j+1n!,where j/η∈N.


## 3.6. Order Statistics

LetX1,X2,…,Xm be a random sample of size m from a distribution with PDF g(x) and CDF G(x) and X1:m,X2:m,…,Xm:m are the analogous order statistics. The PDF and CDF of Xr:m, 1⩽r⩽m, are(17)gr:mx=1Br,m-r+1gxGxr-11-Gxm-r=1Br,m-r+1gx∑i=0m-rmjm-ri-1iGxi+r-1,Gr:m=∑k=rmmkGxk1-Gxm-k,where B(r,m-r+1) is the beta function.Theorem 4.
Letg(x) and G(x) be the PDF and CDF of random variable X~HLGW(ω,η,γ); then the PDF of Xr:m is(18)gr:mx=2ωηγCr:m∑i=0m-rm-ri-1ixη-11+γxηω-1exp⁡1-1+γxηω1+exp⁡1-1+γxηω2×1-exp⁡1-1+γxηω1+exp⁡1-1+γxηωi+r-1,where Cr:m=Br,m-r+1-1.
The CDF corresponding to (18) is (19)Gr:mx=∑k=rmmk1-exp⁡1-1+γxηω1+exp⁡1-1+γxηωk2exp⁡1-1+γxηω1+exp⁡1-1+γxηωm-k.


## 3.7. L-Moments

Suppose that we have a random sampleX1,X2,…,Xn collected from X~HLGW(ω,η,γ). The rth population L-moments are as follows: (20)EXr:n=∫0∞xgXr:ndx=∫0∞x∑i=0n-rn-ri-1i2ωηγCr:nxη-11+γxηω-1e1-1+γxηω1+e1-1+γxηω2×1-e1-1+γxηω1+e1-1+γxηωi+r-1dx.We use the substitution y=1+γxη, so x=y-1/γ1/η and dx=1/γηy-1/γ1/η-1dy, where 1/η∈N. Therefore, we get (21)EXr:n=2ωCr:n∑i=0n-r∑k=0i+r-1∑j=0∞∑l=01/η∑m=0∞n-rii+r-1k-i+r+1j1ηl×-1γ1/η-1j+k+l+mej+k+1j+k+1mm!∫1∞yl+mω+ω-1dy.By working out the integration, we arrive at the following formula:(22)EXr:n=2ω∑i=0n-r∑k=0i+r-1∑j=0∞∑l=01/η∑m=0∞-1γ1/ηj+k+1mej+k+1l+ωm+1m!Aijkl,where (23)Aijkl=Cr:n-1j+k+l+m+1n-rii+r-1k-i+r+1j1ηl.The relation (22) can be used to find out the first L-moments of Xr:n; that is, for r=n=1 we get λ1=E[X1:1]. (24)λ1=2ω∑j=0∞∑m=0∞∑l=01/η-2j1ηl-1γ1/η-1j+l+m+1ej+1j+1ml+ωm+1m!.The other two moments, λ2 and λ3, are, respectively, given by (25)λ2=4ω∑j=0∞∑l=01/η∑m=0∞1ηl-1/γ1/ηl+ωm+1m!×∑i=01∑k=0i1iik-2-ij-1j+k+l+m+1ej+k+lj+k+lm+∑k=011k-3j-1j+k+l+m+1ej+k+1j+k+1m.λ3=2ω∑j=0∞∑l=01/η∑m=0∞1ηl-1/γ1/η-1j+k+l+m+1ej+k+1j+k+1ml+ωm+1m!×∑i=02∑k=0i2iik-2-ij+3!∑i=01∑k=0i+11ii+1k-3-ij+3∑k=022k-4-ij


## 3.8. Quantile Function

ForX to be a random variable with the PDF (6), the quantile function Q(u) is (26)Qu=inf⁡x∈R:Gx⩾u,where0<u<1.The above relation is used to find the quantile function of HLGW distribution. Therefore, we have(27)Qu=1γ1-ln⁡1-u1+u1/ω-11/η.Hence, the generator forX can be given by the following algorithm:(1)
GenerateU~uniform(0,1).(2)
Use (27) and obtain an outcome of X by X=Q(U).By using the quantile function (27), we can examine the Bowley skewness [14] and Moors kurtosis [15] for HLGW as follows: (28)sk=Q3/4+Q1/4-2Q2/4Q3/4-Q1/4κ=Q3/8-Q1/8+Q7/8-Q5/8Q3/4-Q1/4.Table 1 illustrates the values of skewness and kurtosis for the HLGW model for some values of ω, η, and γ. It can be noted that the skewness and kurtosis are free of parameter γ and they are decreasing functions of the parameters ω and η.Table 1
Skewness and kurtosis of HLGW for different values ofω, η, and γ.
γ
ω
η
=
0.5
η
=
1.0
η
=
2.0
s
k
k
u
s
k
k
u
s
k
k
u
0.5
0.5
0.6530
2.4642
0.3366
1.4540
0.1303
1.2492
1.0
0.4634
1.6286
0.1808
1.2395
0.0197
1.2074
1.5
0.3913
1.4463
0.1264
1.1952
−0.0196
1.2045
2.0
0.3541
1.3700
0.0989
1.1779
−0.0396
1.2054
2.5
0.3314
1.3286
0.0824
1.1691
−0.0517
1.2067
1.5
0.5
0.6530
2.4642
0.3366
1.4540
0.1303
1.2492
1.0
0.4634
1.6286
0.1808
1.2395
0.0197
1.2074
1.5
0.3913
1.4463
0.1264
1.1952
−0.0196
1.2045
2.0
0.3541
1.3700
0.0989
1.1779
−0.0396
1.2054
2.5
0.3314
1.3286
0.0824
1.1691
−0.0517
1.2067
2.0
0.5
0.6530
2.4642
0.3366
1.4540
0.1303
1.2492
1.0
0.4634
1.6286
0.1808
1.2395
0.0197
1.2074
1.5
0.3913
1.446
0.1264
1.1952
−0.0196
1.2045
2.0
0.3541
1.3700
0.0989
1.1779
−0.0396
1.2054
2.5
0.3314
1.3286
0.0824
1.1691
−0.0517
1.2067
3.0
0.5
0.6530
2.4642
0.3366
1.4540
0.1303
1.2492
1.0
0.4634
1.6286
0.1808
1.2395
0.0197
1.2074
1.5
0.3913
1.4463
0.1264
1.1956
−0.0196
1.2045
2.0
0.3541
1.3700
0.0989
1.1779
−0.0396
1.2054
2.5
0.3314
1.3286
0.0824
1.1691
−0.0517
1.2067


## 4. Estimation

The log-likelihood function is expressed as(29)lx;ω,η,γ=nln⁡2+nln⁡ω+nln⁡η+nln⁡γ+η-1∑i=1nln⁡xi+ω-1∑i=1nln⁡1+γxiη+∑i=1n1-1+γxiηω-2∑i=1nln⁡1+exp⁡1-1+γxiηω.Taking the first partial derivatives of l(x;ω,η,γ) with respect to ω, η, and γ and letting them equal zero, we obtain a nonlinear system of equations.(30)∂l∂ω=nω+∑i=1nln⁡1+γxiη-∑i=1n1+γxiηωln⁡1+γxiη+2∑i=1n1+γxiηωln⁡1+γxiηe1-1+γxiηω1+e1-1+γxiηω=0.∂l∂η=nη+∑i=1nln⁡xi+ω-1∑i=1nγxiηln⁡xi1+γxiη-ωγ∑i=1nxiηln⁡xi1+γxiηω-1+2ω∑i=1nγxiηln⁡xi1+γxiηω-1e1-1+γxiηω1+e1-1+γxiηω=0∂l∂γ=nγ+ω-1∑i=1nxiη1+γxiη-ω∑i=1n1+γxiηω-1xiη+2ω∑i=1nxiη1+γxiηω-1e1-1+γxiηω1+e1-1+γxiηω=0The above equations cannot be solved analytically, and statistical software can be used to solve them numerically via iterative methods and get the maximum likelihood estimate (MLE) ofω, η, and γ.


### 4.1. Asymptotic Distribution

In order to have approximate confidence intervals (CIs) of the involved parameters, we require the estimated values of the elements of variance-covariance matrixV of the MLEs. The variance-covariance matrix V is estimated by the observed information matrix V^, where(31)V^=-I11I12I13I21I22I23I31I32I33,where Iij,i,j=1,2,3, are the second partial derivatives of (29) with respect to ω, η, and γ. They are the entries of Fisher’s information matrix analogous to ω, η, and γ, respectively, which are given in Appendix. The diagonal of matrix in (31) gives the variances of the MLEs of ω,η, and γ, respectively.Approximation by a standard normal (SN) distribution of the distribution ofZθ^k=(θ^k-θk)/Var^(θ^k), where θ^=(ω^,η^,γ^), results in an approximate 100(1-ϑ)% confidence interval for θk as (32)θ^k±zϑ/2Var^θ^k,j=1,2,3,where zϑ/2 is the upper (ϑ/2)100th percentile of SN distribution. We can use the likelihood ratio (LR) test to compare the fit of the HLGW distribution with its submodels for a given data set. For example, to test γ=0, the LR statistic is w=2ln⁡(L(ω^,η^,γ^))-ln⁡Lω~,η~,0, where ω^,η^, and γ^ are the unrestricted estimates and ω~,η~ are the restricted estimates. The LR test rejects the null hypothesis if w>χϵ2, where χϵ2 denotes the upper 100∈% point of the χ2-distribution with 1 degree of freedom.


## 4.1. Asymptotic Distribution

In order to have approximate confidence intervals (CIs) of the involved parameters, we require the estimated values of the elements of variance-covariance matrixV of the MLEs. The variance-covariance matrix V is estimated by the observed information matrix V^, where(31)V^=-I11I12I13I21I22I23I31I32I33,where Iij,i,j=1,2,3, are the second partial derivatives of (29) with respect to ω, η, and γ. They are the entries of Fisher’s information matrix analogous to ω, η, and γ, respectively, which are given in Appendix. The diagonal of matrix in (31) gives the variances of the MLEs of ω,η, and γ, respectively.Approximation by a standard normal (SN) distribution of the distribution ofZθ^k=(θ^k-θk)/Var^(θ^k), where θ^=(ω^,η^,γ^), results in an approximate 100(1-ϑ)% confidence interval for θk as (32)θ^k±zϑ/2Var^θ^k,j=1,2,3,where zϑ/2 is the upper (ϑ/2)100th percentile of SN distribution. We can use the likelihood ratio (LR) test to compare the fit of the HLGW distribution with its submodels for a given data set. For example, to test γ=0, the LR statistic is w=2ln⁡(L(ω^,η^,γ^))-ln⁡Lω~,η~,0, where ω^,η^, and γ^ are the unrestricted estimates and ω~,η~ are the restricted estimates. The LR test rejects the null hypothesis if w>χϵ2, where χϵ2 denotes the upper 100∈% point of the χ2-distribution with 1 degree of freedom.


## 5. Simulation Study

The MLEs can be checked out by a simulation study. The following steps can be followed:(1)
By using (6), 5,000 samples of size n are achieved. The variates of the HLGW distribution are developed using (33)X=1γ1-ln⁡1-u1+u1/ω-11/η,foru~U(0,1).(2)
The MLEs are computed for the samples, sayΘ^j=(ω^j,η^j,γ^j) for j=1,2,…,5,000.(3)
The mean square errors (MSEs) are calculated for every parameter.The above steps were repeated forn=20,40,60,80,100,250, and 500 with ω=1.5,η=5,γ=0.5 and ω=1.5,η=10,γ=1. Table 2 shows the bias and MSEs of ω,η, and γ. It can be deduced through the table that MSEs for individual parameters fall to zero when sample size increases.Table 2
Bias and MSE for the HLGW parameters.
ω
η
γ
n
Bias (ω)
MSE (ω)
Bias (η)
MSE (η)
Bias (γ)
MSE (γ)
1.5
5
0.5
20
0.2103
0.0392
1.1935
3.2415
0.3454
0.9104
40
0.1732
0.0314
1.1431
2.9381
0.2931
0.5830
60
0.1526
0.0283
1.1063
2.4897
0.2480
0.3148
80
0.1473
0.0231
0.9733
2.1738
0.1812
0.1908
100
0.1332
0.0207
0.8910
1.9318
0.1317
0.0813
250
0.1171
0.0113
0.5337
1.3877
0.0811
0.0031
500
0.1010
0.0061
0.2701
1.0814
0.0213
0.0008
1.5
10
1
20
0.3118
0.0433
2.6754
15.9312
0.6125
10.2311
40
0.2918
0.0395
2.3487
12.3471
0.2451
7.6401
60
0.2554
0.0365
2.1174
9.2877
0.1615
4.3881
80
0.2375
0.0335
2.0968
6.2514
0.0941
1.9722
100
0.2114
0.0317
2.0532
4.9934
0.0532
1.5531
250
0.1783
0.0307
0.0065
1.9951
0.0113
0.4899
500
0.1265
0.0299
0.0013
0.2164
0.0095
0.0989


## 6. Applications

This section deals with the applications of the HLGW model to two lifetime data sets, that is, the data of 213 observed values of intervals between failures of air conditioning system of Boeing 720 jet airplanes and the data of waiting times (min) of 100 bank customers. Estimates of the parameters of HLGW distribution (standard errors in parentheses) and Cramer-von MisesW∗, Anderson Darling A∗, and K-S statistics are presented for the data sets. In general, the smaller the values of W∗, A∗, and K-S statistics, the better the fit. We compare the proposed model with other models for the same data sets to check the potentiality and flexibility of new model.


### 6.1. Air Conditioning Systems Failure Data

The first application concerns 213 observed values of intervals between failures of air conditioning system of Boeing 720 jet airplanes firstly analysed by Proschan [16]. We have compared the performance of the HLGW distribution with its submodels as well as with some other well-known models given below.The Weibull Poisson (WP) distribution by Lu and Shi [17] with the PDF: (34)gt;α,β,λ=αβλtα-11-e-λe-λ-βtα+λexp⁡-βtα,t>0,for α, β, and λ>0.The complementary Weibull geometric (CWG) distribution by Tojeiro et al. [18] with the PDF: (35)gt=γλγθtγ-1e-λtγθ+1-θe-λtγ2,for t,λ,γ, and θ>0.The Weibull (W) distribution with PDF:(36)gy;β,γ=βγyγ-1exp⁡-βyγ,for y>0, β>0, and γ>0.The estimated values of the parameters with standard errors (SE) are found using the method of maximum likelihood estimation. Table3 gives the estimated values of the parameters along with their standard errors and the test statistics have the smallest values of W∗, A∗, and K-S statistic for the data set under HLGW distribution as compared to the other models. Based on these values, it is concluded that HLGW distribution is the best model as compared to the other models to fit this data set. This conclusion can also be made by the CDF plots for empirical and fitted HLGW distributions using data of 213 values of intervals between failures of air conditioning system in Figure 3(a) for the data. In Figure 3(b), the TTT plot is shown for the data set. The failure rate shape for this data set is decreasing, as its TTT plot is convex.Table 3
Estimates of models for the air conditioning systems failure data.
Distributions
Estimates
-
2
l
(
θ
^
)
Statistics
ω
^
η
^
γ
^
W
∗
A
∗
K-S
p value
HLGW
0.3470
1.3814
0.0296
2349.674
0.0339
0.2587
0.0394
0.8957
(0.0885)
(0.2462)
(0.0127)
HLW
0.7804
0.0442
2360.630
0.2099
1.2851
0.0577
0.4770
(0.0426)
(0.0098)
HLE
0.0144
2383.325
0.3446
2.0606
0.1407
0.0004
(0.0009)
HLNH
0.5596
0.0482
2352.930
0.0850
0.5534
0.0445
0.7926
(0.0515)
(0.0108)
Weibull
0.0158
0.9226
2355.171
0.1373
0.8552
0.0514
0.6270
(0.0038)
(0.0459)
CWG
0.2890
0.0079
4.1050
2362.329
0.1519
1.0253
0.0502
0.6563
(0.0238)
(0.0032)
(2.8847)
WP
0.4015
0.4967
−8.0978
2351.850
0.0440
0.3305
0.0412
0.8624
(0.0584)
(0.1838)
(2.5766)Figure 3
TTT plot and CDF plot for the air conditioning systems failure data.
(a)
TTT plot
(b)
CDF PlotThe LR test statistics of hypothesesH0: HLW versus Ha: HLGW, H0: HLE versus Ha: HLGW, and H0: HLNH versus Ha: HLGW are 10.956 (p value = 0.00093), 33.651 (p value = 0.00001), and 3.256 (p value = 0.071163), respectively. We conclude that there is a significant difference between HLW and HLGW distributions, HLE and HLGW, and also between HLNH and HLGW distributions at the 10% level.


### 6.2. Waiting Time Data

The data encountered in the second application involves data of waiting times (min) of 100 bank customers used by Bidram and Nadarajah [4]. We have compared the performance of the HLGW distribution with its submodels as well as with some other well-known distributions, such as the Weibull and the complementary Weibull geometric distribution. We also use the LR test to compare the HLGW distribution and its submodels.The estimated values of the parameters, the standard errors, and the goodness-of-fit test statisticsW∗, A∗, and K-S statistic for the data of waiting time are given in Table 4. These tables illustrate that HLGW model shows a good fit for this data set as compared to the other distributions. The CDF plot for empirical and HLGW distributions using the data set of waiting time in Figure 4(a) also confirms the fitness of HLGW distribution. Figure 4(b) expresses the TTT plot for this data set. Since the TTT plot of the data set is concave, the data set has increasing hazard rate shape. The LR test statistics of the hypotheses H0: HLW versus Ha: HLGW, H0: HLE versus Ha: HLGW, and H0: HLNH versus Ha: HLGW are 5.501 (p value = 0.019006), 11.85 (p value = 0.000577), and 9.271 (p value = 0.002328). The HLGW distribution is significantly better than HLW, HLE, and HLNH distributions. There is no difference between HLGW and Weibull distribution based on the LR test; however the goodness-of-fit statistics W∗, A∗, and K-S statistic clearly show that HLGW distribution is better than Weibull distribution for the data.Table 4
Estimates of models for the waiting time data.
Distributions
Estimates
-
2
l
(
θ
^
)
Statistics
ω
^
η
^
γ
^
W
∗
A
∗
K-S
p value
HLGW
0.3610
2.1425
0.0773
634.226
0.0176
0.1277
0.0368
0.9992
(0.1268)
(0.5257)
(0.0340)
HLW
1.2336
0.0772
639.730
0.0934
0.5740
0.0627
0.8267
(0.0978)
(0.0216)
HLE
0.1446
646.076
0.0583
0.3624
0.1213
0.1053
(0.0119)
HLNH
1.4044
0.0846
643.497
0.1032
0.6355
0.0877
0.4257
(0.3411)
(0.0316)
Weibull
0.0305
1.4582
637.461
0.0629
0.3961
0.0577
0.8938
(0.0095)
(0.1089)
CWG
0.5066
0.0126
2.2254
645.742
0.0745
0.4653
0.0485
0.9726
(0.0779)
(0.0097)
(1.7142)Figure 4
TTT plot and CDF plot for the waiting time data.
(a)
TTT plot
(b)
CDF Plot


## 6.1. Air Conditioning Systems Failure Data

The first application concerns 213 observed values of intervals between failures of air conditioning system of Boeing 720 jet airplanes firstly analysed by Proschan [16]. We have compared the performance of the HLGW distribution with its submodels as well as with some other well-known models given below.The Weibull Poisson (WP) distribution by Lu and Shi [17] with the PDF: (34)gt;α,β,λ=αβλtα-11-e-λe-λ-βtα+λexp⁡-βtα,t>0,for α, β, and λ>0.The complementary Weibull geometric (CWG) distribution by Tojeiro et al. [18] with the PDF: (35)gt=γλγθtγ-1e-λtγθ+1-θe-λtγ2,for t,λ,γ, and θ>0.The Weibull (W) distribution with PDF:(36)gy;β,γ=βγyγ-1exp⁡-βyγ,for y>0, β>0, and γ>0.The estimated values of the parameters with standard errors (SE) are found using the method of maximum likelihood estimation. Table3 gives the estimated values of the parameters along with their standard errors and the test statistics have the smallest values of W∗, A∗, and K-S statistic for the data set under HLGW distribution as compared to the other models. Based on these values, it is concluded that HLGW distribution is the best model as compared to the other models to fit this data set. This conclusion can also be made by the CDF plots for empirical and fitted HLGW distributions using data of 213 values of intervals between failures of air conditioning system in Figure 3(a) for the data. In Figure 3(b), the TTT plot is shown for the data set. The failure rate shape for this data set is decreasing, as its TTT plot is convex.Table 3
Estimates of models for the air conditioning systems failure data.
Distributions
Estimates
-
2
l
(
θ
^
)
Statistics
ω
^
η
^
γ
^
W
∗
A
∗
K-S
p value
HLGW
0.3470
1.3814
0.0296
2349.674
0.0339
0.2587
0.0394
0.8957
(0.0885)
(0.2462)
(0.0127)
HLW
0.7804
0.0442
2360.630
0.2099
1.2851
0.0577
0.4770
(0.0426)
(0.0098)
HLE
0.0144
2383.325
0.3446
2.0606
0.1407
0.0004
(0.0009)
HLNH
0.5596
0.0482
2352.930
0.0850
0.5534
0.0445
0.7926
(0.0515)
(0.0108)
Weibull
0.0158
0.9226
2355.171
0.1373
0.8552
0.0514
0.6270
(0.0038)
(0.0459)
CWG
0.2890
0.0079
4.1050
2362.329
0.1519
1.0253
0.0502
0.6563
(0.0238)
(0.0032)
(2.8847)
WP
0.4015
0.4967
−8.0978
2351.850
0.0440
0.3305
0.0412
0.8624
(0.0584)
(0.1838)
(2.5766)Figure 3
TTT plot and CDF plot for the air conditioning systems failure data.
(a)
TTT plot
(b)
CDF PlotThe LR test statistics of hypothesesH0: HLW versus Ha: HLGW, H0: HLE versus Ha: HLGW, and H0: HLNH versus Ha: HLGW are 10.956 (p value = 0.00093), 33.651 (p value = 0.00001), and 3.256 (p value = 0.071163), respectively. We conclude that there is a significant difference between HLW and HLGW distributions, HLE and HLGW, and also between HLNH and HLGW distributions at the 10% level.


## 6.2. Waiting Time Data

The data encountered in the second application involves data of waiting times (min) of 100 bank customers used by Bidram and Nadarajah [4]. We have compared the performance of the HLGW distribution with its submodels as well as with some other well-known distributions, such as the Weibull and the complementary Weibull geometric distribution. We also use the LR test to compare the HLGW distribution and its submodels.The estimated values of the parameters, the standard errors, and the goodness-of-fit test statisticsW∗, A∗, and K-S statistic for the data of waiting time are given in Table 4. These tables illustrate that HLGW model shows a good fit for this data set as compared to the other distributions. The CDF plot for empirical and HLGW distributions using the data set of waiting time in Figure 4(a) also confirms the fitness of HLGW distribution. Figure 4(b) expresses the TTT plot for this data set. Since the TTT plot of the data set is concave, the data set has increasing hazard rate shape. The LR test statistics of the hypotheses H0: HLW versus Ha: HLGW, H0: HLE versus Ha: HLGW, and H0: HLNH versus Ha: HLGW are 5.501 (p value = 0.019006), 11.85 (p value = 0.000577), and 9.271 (p value = 0.002328). The HLGW distribution is significantly better than HLW, HLE, and HLNH distributions. There is no difference between HLGW and Weibull distribution based on the LR test; however the goodness-of-fit statistics W∗, A∗, and K-S statistic clearly show that HLGW distribution is better than Weibull distribution for the data.Table 4
Estimates of models for the waiting time data.
Distributions
Estimates
-
2
l
(
θ
^
)
Statistics
ω
^
η
^
γ
^
W
∗
A
∗
K-S
p value
HLGW
0.3610
2.1425
0.0773
634.226
0.0176
0.1277
0.0368
0.9992
(0.1268)
(0.5257)
(0.0340)
HLW
1.2336
0.0772
639.730
0.0934
0.5740
0.0627
0.8267
(0.0978)
(0.0216)
HLE
0.1446
646.076
0.0583
0.3624
0.1213
0.1053
(0.0119)
HLNH
1.4044
0.0846
643.497
0.1032
0.6355
0.0877
0.4257
(0.3411)
(0.0316)
Weibull
0.0305
1.4582
637.461
0.0629
0.3961
0.0577
0.8938
(0.0095)
(0.1089)
CWG
0.5066
0.0126
2.2254
645.742
0.0745
0.4653
0.0485
0.9726
(0.0779)
(0.0097)
(1.7142)Figure 4
TTT plot and CDF plot for the waiting time data.
(a)
TTT plot
(b)
CDF Plot


## 7. Conclusion

We have proposed a three-parameter lifetime generalized distribution, referred to as the half-logistic generalized Weibull (HLGW) distribution. The HLGW distribution has three other distributions like the half-logistic exponential, the half-logistic Weibull, and the half-logistic Nadarajah-Haghighi distributions, as its submodels. The new model exhibits a variety of shapes of the failure rate function, that is, increasing, increasing and then constant, decreasing, bathtub, unimodal, and decreasing-increasing-decreasing (DID) shapes. Various statistical properties of the HLGW distribution are derived and studied in detail. We estimated the parameters involved in the model by using the method of MLEs. Two real data sets are used to illustrate the flexibility, potentiality, and usefulness of HLGW distribution. It is concluded that HLGW model delivers better fitting than the other lifetime models and we hope that HLGW distribution may attract wider range of practical applications and this research may serve as a reference and benefit future research in the subject field of study.



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*Source: 8767826-2018-01-04.xml*