# Harmonic Mixture Weibull-G Family of Distributions: Properties, Regression and Applications to Medical Data

Authors: Ernest Zamanah; Suleman Nasiru; Albert Luguterah
Journal: Computational and Mathematical Methods
 (2022)

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## Abstract

In recent years, the developments of new families of probability distributions have received greater attention as a result of desirable properties they exhibit in the modelling of data sets. The Harmonic Mixture Weibull-G family of distributions was developed in this study. The statistical properties were comprehensively presented and five special distributions developed from the family. The hazard functions of the special distributions were shown to exhibit various forms of monotone and nonmonotone shapes. The applications of the developed family to real data sets in medical studies revealed that the special distribution (Harmonic mixture Weibul Weibull distribution) provided a better fit to the data sets than other competitive models. A location-scale regression model was developed from the family and its application demonstrated using survival time data of hypertensive patients.


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## Body

## 1. Introduction

Advances in the field of medicine are critical to the well-being of humankind. To this end, the need for the use of appropriate and very efficient probability distributions in the modelling of medical data is fundamentally important. The efficient modelling of medical data is useful in providing good understanding of the distribution of disease incidence and prevalence in medical studies.In medical and biological studies, several phenotypic traits including chronic conditions such as cancer, diabetes, hypertension, and cardiovascular diseases among others are usually encountered. Appropriate knowledge about the distribution of disease incidence and prevalence in a population enhances the development of appropriate hypotheses about underlying mechanisms of health and disease [1]. This is profoundly important in advancing the course of medicine.In medical and biological studies, the Weibull distribution among numerous classical distributions is a widely applied model for analyzing data with monotone hazard rate shapes. For complex biological phenotypic traits with nonmonotone hazard rate shapes, the Weibull distribution does not have the flexibility to model such data. Consequently, new families of distributions in the form of extended or modified versions of the Weibull distribution have been introduced in literature with the attempt of increasing its flexibility. Some examples include the following: Marshall-Olkin Weibull generated family [2], exponentiated power generalized Weibull power series family of distributions [3], complementary generalized power Weibull power series family of distributions [4], the Burr-Weibull power series family [5], extended Weibull-G family [6], Weibull Burr X–G family of distributions [7], the Weibull Marshall–Olkin family [8], the gamma-Weibull-G family [9], generalized odd Weibull generated family [10], the beta Weibull-G family [11], Kumaraswamy Weibull-generated family [12], generalized extended Weibull power series family of distributions [13], the inverse Weibull power series family [14], the Marshall-Olkin extended Weibull family of distributions, [15] and the extended Weibull power series family [16].In this study, we proposed a novel generalization for the Weibull-G family, called the Harmonic mixture Weibull-G family by combining the Harmonic mixture-G [17] and the Weibull-G [18] families. The major motivations behind generating this family include the following: to develop special distributions capable of modelling medical data that are characterized with bimodality; to generate distributions with the capability of modelling medical data that are characterized with both monotone and nonmonotone hazard rate shapes; to produce special distributions that can generalize some well-known models in the literature; to generate more flexible distributions that take into consideration skewness, kurtosis, and tail variations in the modelling of medical data; to develop alternative distributions with superior parametric fits to data in medical studies than existing classical distributions; and to develop a location-scale regression model for studying the relationship between a response variable and a set of covariates.The remainder of the article is structured as follows: Section2 presents the development of the new family. Section 3 presents some statistical properties of the family. Section 4 presents the maximum likelihood estimation of the parameters. Section 5 presents some special distributions. Section 6 presents the location-scale regression model. Section 7 presents simulation results. Section 8 presents the applications of the developed family. Section 9 finally presents the conclusions of the study.


## 2. Development of the Harmonic Mixture Weibull-G Family

Suppose that the continuous random variableX follows the Weibull-G family of distributions. Then, according to Bourguignon et al. [18], the cumulative distribution function (CDF) and probability density function (PDF) are, respectively, given by
(1)Fx=1−exp−αGx;ξG¯x;ξβ,x∈D⊆ℝ;α,β>0,and
(2)fx;α,β,ξ=αβgx;ξGx;ξβ−1G¯x;ξβ+1exp−αGx;ξG¯x;ξβ.If the random variableX follows the Harmonic mixture-G (HM-G) family, then according to Kharazmi et al. [17], the CDF and PDF are, respectively, given by
(3)Hx=1−G¯αx1−θ1−G¯α−1x,x∈ℝ,α≥0,0<θ<1,and
(4)hx=gxG¯α−1xα1−θ+θG¯α−1x1−θ1−G¯α−1x2,x∈ℝ.The Harmonic mixture Weibull generated (HMW-G) family of distributions is developed in this section by combining the CDFs of the HM-G and Weibull-G families. Suppose that the random variableX follows the HMW-G family of distributions, the CDF of the HMW-G family is given by
(5)FXx=1−exp−αGx;φ/G¯x;φβ1−θ1−exp−α−1Gx;φ/G¯x;φβ,x∈ℝ,where α>0 and 0<θ<1 are scale parameters, β>0 is a shape parameter, and φ is a p×1 vector of parameters. When α=0, the HMW-G family reduces to the Marshall-Olkin Weibull-G family of distributions. The corresponding PDF of the HMW-G family is the first derivative of its CDF. Thus, the PDF is given by
(6)fXx=βgx;φGx;φβ−1G¯x;φβ+1exp−αGx;φG¯x;φβα1−θ+θexp−α−1Gx;φ/G¯x;φβ1−θ1−exp−α−1Gx;φ/G¯x;φβ2,x∈ℝ.The hazard rate function of the family is given by(7)rXx=βgx;φGx;φβ−1G¯x;φβ+1α1−θ+θexp−α−1Gx;φ/G¯x;φβ1−θ1−exp−α−1Gx;φ/G¯x;φβ,x∈ℝ.Lemma 1.
The mixture representation of the density function of the HMW-G family is(8)fXx=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmgx;φGx;φβk+1+m−1,where
(9)ω¯ijkm=−1j+ki+1α+jα−1kθiα1−θk!βk+1+mm,ωijkm=−1j+ki+1θi+12α+jα−1−1kk!βk+1+mm.Proof.
Using the binomial series expansion1−z−n=∑i=0∞n+i−1izi,z<1,(10)1−θ1−e−α−1Gx;φG¯x;φβ−2=∑i=0∞∑j=0i−1ii+1θie−jα−1Gx;φG¯x;φβ.
Thus, the PDF of the HMW-G family can be written as(11)fXx=βgx;φGx;φβ−1G¯x;φβ+1∑i=0∞∑j=0i−1ji+1θiα1−θe−α+jα−1Gx;φG¯x;φβ+βgx;φGx;φβ−1G¯x;φβ+1∑i=0∞∑j=0i−1ji+1θi+1e−2α+jα−1−1Gx;φG¯x;φβ.
Using the Taylor series, we have(12)fXx=βgx;φ∑i=0∞∑j=0i∑k=0∞−1j+ki+1α+jα−1kθiα1−θk!Gx;φβk+1−1G¯x;φβk+1+1+βgx;φ∑i=0∞∑j=0i∑k=0∞−1j+ki+12α+jα−1−1kθi+1k!Gx;φβk+1−1G¯x;φβk+1+1.
Applying the binomial series expansion,(13)fXx=βgx;φ∑i=0∞∑j=0i∑k=0∞∑m=0∞−1j+ki+1α+jα−1kθiα1−θk!βk+1+mmGx;φβk+1+m−1+βgx;φ∑i=0∞∑j=0i∑k=0∞∑m=0∞−1j+ki+12α+jα−1−1kθi+1k!βk+1+mmGx;φβk+1+m−1.
Thus,(14)fXx=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmgx;φGx;φβk+1+m−1.
This completes the proof.


## 3. Statistical Properties

In this section, statistical properties of the HMW-G family of distributions are presented.


### 3.1. Quantile Function

The quantile function plays an important role in simulating random samples from a given distribution. For a given distribution, the characteristics such as median, kurtosis, and skewness can also be described using the quantile function.Proposition 2.
The quantile function of the HMW-G family foru∈0,1 is given by
(15)1−u1−θexp−α−1Gx;φG¯x;φβ−exp−αGx;φG¯x;φβ=0.Proof.
Using the CDF of the HMW-G family defined in equation (5), let U be a random variable having the uniform distribution on the interval [0, 1]. Then, the uth quantile, denoted by xu is obtained such that,
(16)1−exp−αGx;φ/G¯x;φβ1−θ1−exp−α−1Gx;φ/G¯x;φβ=u.
This implies that(17)1−θ1−exp−α−1Gx;φG¯x;φβ−exp−αGx;φG¯x;φβ=u1−θ1−exp−α−1Gx;φG¯x;φβ.
Hence, the quantile is obtained as the solution of(18)1−u1−θ1−exp−α−1Gx;φG¯x;φβ−exp−αGx;φG¯x;φβ=0.
This completes the proof.


### 3.2. Moments

In this section, the expression for therth order moment of the HMW-G family of distributions is derived. It can be used to compute measures of dispersion, kurtosis, and skewness of data sets in medical studies.Proposition 3.
Therth noncentral moment of the HMW-G family of distributions is
(19)μr′=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞∞xrgx;φGx;φβk+1+m−1dx,r=1,2,.⋯Proof.
Therth noncentral moment is defined as
(20)μr′=∫−∞∞xrfXxdx.
Substituting the mixture representation of the density function into the definition, we have(21)μr′=∫−∞∞xrβ∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmgx;φGx;φβk+1+m−1dx.
This implies that(22)μr′=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞∞xrgx;φGx;φβk+1+m−1dx,r=1,2,.⋯
This completes the proof.


### 3.3. Incomplete Moment

In this section, the expression for therth incomplete moment of the HMW-G family of distributions is derived. It can be used to determine the mean deviation or median deviation of data sets in medical studies.Proposition 4.
Therth incomplete moment of the HMW-G family of distributions is
(23)ψry=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞yxrgx;φGx;φβk+1+m−1dx,r=1,2,.⋯Proof.
By definition, therth incomplete moment is given by
(24)ψry=EXX<Y=∫−∞yxrfXxdx.
Substituting the mixture representation of the density function into the definition, we have(25)ψry=∫−∞yxrβ∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmgx;φGx;φβk+1+m−1dx.
This implies that(26)ψry=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞yxrgx;φGx;φβk+1+m−1dx,r=1,2,.⋯


### 3.4. Moment Generating Function

In this section, the expression for the moment generating function (MGF) of the HMW-G family of distributions is presented. The MGF is useful in finding the moments of a random variable. The MGF of a random variableX having the HMW-G distribution if it exists is given by the following proposition.Proposition 5.
The MGF of the HMW-G family of distribution is given by(27)MXt=β∑r=0∞∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmtrr!∫−∞∞gx;φGx;φβk+1+m−1dx,r=1,2,.⋯Proof.
By definition, the MGF is given by(28)MXt=EetX=∫−∞∞etxfXxdx.
Using the Taylor series expansion,(29)etX=∑r=0∞trXrr!.
This implies that(30)MXt=E∑r=0∞trXrr!=∑r=0∞trr!EXr=∑r=0∞trr!μr′.
Substitutingμr′ into equation (30) gives
(31)MXt=β∑r=0∞∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmtrr!∫−∞∞gx;φGx;φβk+1+m−1dx,r=1,2,.⋯
This completes the proof.


### 3.5. Mean Residual Life

The Mean Residual Life (MRL) functionmy, is a function that characterizes the distribution function Fy, uniquely [19]. It describes the average survival time of a component after it exceeds a specific timey. The MRL function plays a key role in survival analysis when analyzing the event time of a given phenotypic trait in medical studies.Proposition 6.
IfY is a random variable representing the life time of a component with distribution function Fy, then the MRL of the HMW-G family of distributions is
(32)my=μ−β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞yxgx;φGx;φβk+1+m−1dx1−FXy−y,where μ=μ1′..Proof.
By definition, the MRL is given by(33)my=EX−yX>y=11−FXy∫y∞x−yfxdx=μ1′−∫−∞yxfXxdx1−FXy−y.
Hence, substituting the first incomplete moment∫−∞yxfXxdx, into equation (33) gives
(34)my=μ−β∑i=0∞∑i=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞y∫−∞yxgx;φGx;φβk+1+m−1dx1−FXy−y.
This completes the proof.


### 3.6. Identifiability

The identifiability property of the HMW-G family is studied in this section. The identifiability property of the model is essential to ensure that precise inferences are possible.Proposition 7.
LetX1 be HMW-G family random variable with CDF FXx;α1,β1,θ1,φ1 and X2 be HMW-G family random variable with CDF FXx;α2,β2,θ2,φ2. Then, the HMW-G family is identifiable if α1=α2,β1=β2,θ1=θ2 and φ1=φ2.Proof.
For the HMW-G family of distributions to be identifiable,FXx;α1,β1,θ1,φ1=FXx;α2,β2,θ2,φ2.
Hence,(35)1−exp−α1Gx;φ1/G¯x;φ1β11−θ11−exp−α1−1Gx;φ1/G¯x;φ1β1=1−exp−α2Gx;φ2/G¯x;φ2β21−θ21−exp−α2−1Gx;φ2/G¯x;φ2β2.
Ifα1=α2,β1=β2,θ1=θ2 and φ1=φ2,
(36)exp−α2Gx;φ2/G¯x;φ2β21−θ21−exp−α2−1Gx;φ2/G¯x;φ2β2−exp−α1Gx;φ1/G¯x;φ1β11−θ11−exp−α1−1Gx;φ1/G¯x;φ1β1=0.
Hence, the identifiability condition is satisfied. This completes the proof.


## 3.1. Quantile Function

The quantile function plays an important role in simulating random samples from a given distribution. For a given distribution, the characteristics such as median, kurtosis, and skewness can also be described using the quantile function.Proposition 2.
The quantile function of the HMW-G family foru∈0,1 is given by
(15)1−u1−θexp−α−1Gx;φG¯x;φβ−exp−αGx;φG¯x;φβ=0.Proof.
Using the CDF of the HMW-G family defined in equation (5), let U be a random variable having the uniform distribution on the interval [0, 1]. Then, the uth quantile, denoted by xu is obtained such that,
(16)1−exp−αGx;φ/G¯x;φβ1−θ1−exp−α−1Gx;φ/G¯x;φβ=u.
This implies that(17)1−θ1−exp−α−1Gx;φG¯x;φβ−exp−αGx;φG¯x;φβ=u1−θ1−exp−α−1Gx;φG¯x;φβ.
Hence, the quantile is obtained as the solution of(18)1−u1−θ1−exp−α−1Gx;φG¯x;φβ−exp−αGx;φG¯x;φβ=0.
This completes the proof.


## 3.2. Moments

In this section, the expression for therth order moment of the HMW-G family of distributions is derived. It can be used to compute measures of dispersion, kurtosis, and skewness of data sets in medical studies.Proposition 3.
Therth noncentral moment of the HMW-G family of distributions is
(19)μr′=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞∞xrgx;φGx;φβk+1+m−1dx,r=1,2,.⋯Proof.
Therth noncentral moment is defined as
(20)μr′=∫−∞∞xrfXxdx.
Substituting the mixture representation of the density function into the definition, we have(21)μr′=∫−∞∞xrβ∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmgx;φGx;φβk+1+m−1dx.
This implies that(22)μr′=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞∞xrgx;φGx;φβk+1+m−1dx,r=1,2,.⋯
This completes the proof.


## 3.3. Incomplete Moment

In this section, the expression for therth incomplete moment of the HMW-G family of distributions is derived. It can be used to determine the mean deviation or median deviation of data sets in medical studies.Proposition 4.
Therth incomplete moment of the HMW-G family of distributions is
(23)ψry=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞yxrgx;φGx;φβk+1+m−1dx,r=1,2,.⋯Proof.
By definition, therth incomplete moment is given by
(24)ψry=EXX<Y=∫−∞yxrfXxdx.
Substituting the mixture representation of the density function into the definition, we have(25)ψry=∫−∞yxrβ∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmgx;φGx;φβk+1+m−1dx.
This implies that(26)ψry=β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞yxrgx;φGx;φβk+1+m−1dx,r=1,2,.⋯


## 3.4. Moment Generating Function

In this section, the expression for the moment generating function (MGF) of the HMW-G family of distributions is presented. The MGF is useful in finding the moments of a random variable. The MGF of a random variableX having the HMW-G distribution if it exists is given by the following proposition.Proposition 5.
The MGF of the HMW-G family of distribution is given by(27)MXt=β∑r=0∞∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmtrr!∫−∞∞gx;φGx;φβk+1+m−1dx,r=1,2,.⋯Proof.
By definition, the MGF is given by(28)MXt=EetX=∫−∞∞etxfXxdx.
Using the Taylor series expansion,(29)etX=∑r=0∞trXrr!.
This implies that(30)MXt=E∑r=0∞trXrr!=∑r=0∞trr!EXr=∑r=0∞trr!μr′.
Substitutingμr′ into equation (30) gives
(31)MXt=β∑r=0∞∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkmtrr!∫−∞∞gx;φGx;φβk+1+m−1dx,r=1,2,.⋯
This completes the proof.


## 3.5. Mean Residual Life

The Mean Residual Life (MRL) functionmy, is a function that characterizes the distribution function Fy, uniquely [19]. It describes the average survival time of a component after it exceeds a specific timey. The MRL function plays a key role in survival analysis when analyzing the event time of a given phenotypic trait in medical studies.Proposition 6.
IfY is a random variable representing the life time of a component with distribution function Fy, then the MRL of the HMW-G family of distributions is
(32)my=μ−β∑i=0∞∑j=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞yxgx;φGx;φβk+1+m−1dx1−FXy−y,where μ=μ1′..Proof.
By definition, the MRL is given by(33)my=EX−yX>y=11−FXy∫y∞x−yfxdx=μ1′−∫−∞yxfXxdx1−FXy−y.
Hence, substituting the first incomplete moment∫−∞yxfXxdx, into equation (33) gives
(34)my=μ−β∑i=0∞∑i=0i∑k=0∞∑m=0∞ω¯ijkm+ωijkm∫−∞y∫−∞yxgx;φGx;φβk+1+m−1dx1−FXy−y.
This completes the proof.


## 3.6. Identifiability

The identifiability property of the HMW-G family is studied in this section. The identifiability property of the model is essential to ensure that precise inferences are possible.Proposition 7.
LetX1 be HMW-G family random variable with CDF FXx;α1,β1,θ1,φ1 and X2 be HMW-G family random variable with CDF FXx;α2,β2,θ2,φ2. Then, the HMW-G family is identifiable if α1=α2,β1=β2,θ1=θ2 and φ1=φ2.Proof.
For the HMW-G family of distributions to be identifiable,FXx;α1,β1,θ1,φ1=FXx;α2,β2,θ2,φ2.
Hence,(35)1−exp−α1Gx;φ1/G¯x;φ1β11−θ11−exp−α1−1Gx;φ1/G¯x;φ1β1=1−exp−α2Gx;φ2/G¯x;φ2β21−θ21−exp−α2−1Gx;φ2/G¯x;φ2β2.
Ifα1=α2,β1=β2,θ1=θ2 and φ1=φ2,
(36)exp−α2Gx;φ2/G¯x;φ2β21−θ21−exp−α2−1Gx;φ2/G¯x;φ2β2−exp−α1Gx;φ1/G¯x;φ1β11−θ11−exp−α1−1Gx;φ1/G¯x;φ1β1=0.
Hence, the identifiability condition is satisfied. This completes the proof.


## 4. Parameter Estimation

In this section, the maximum likelihood estimation (MLE) procedure is presented for the estimation of the unknown parameters of the HMW-G family. Letx1,x2,⋯,xn be a random sample of size n from the HMW-G family of distributions with Ψ=α,β,ϕ,φ′ an unknown parameter vector where φ is a p×1 parameter vector for the baseline distribution. Under these settings, the total log-likelihood function is
(37)ℓ=nlogβ+∑i=1nloggxi;φ+β−1∑i=1nlogGxi;φ−β+1∑i=1nlogG¯xi;φ−α∑i=1nGx;φG¯x;φβ+∑i=1nlogα1−θ+θexp−α−1Gx;φG¯x;φβ−2∑i=1nlog1−θ1−exp−α−1Gx;φG¯x;φβ.The score vectorsUΨ of the likelihood function are obtained by taking partial derivatives of (35) with respect to the parameters α,β,θandφ as
(38)UΨ=∂ℓ∂Ψ=∂ℓ∂α,∂ℓ∂β,∂ℓ∂θ,∂ℓ∂φ′.By setting the score vectors to zero, the simultaneous solution of the system of nonlinear equations gives the maximum likelihood estimates of the parameters. However, this nonlinear system of equations does not have a closed form. Thus, we apply numerical optimization to maximize the log-likelihood function directly using R software.


## 5. Special Distributions

In this section, some special cases of the HMW-G family of distributions are developed and studied.


### 5.1. Harmonic Mixture Weibull Burr III Distribution

Suppose that the baseline model of the HMW-G family is the Burr III distribution with CDF and PDF, respectively, defined by Burr [20] as Gx=1+x−a−b and gx=abx−a−11+x−a−b−1,x>0,a>0,b>0. Then, the PDF and hazard function of the HMW-Burr III (HMWBIII) distribution are given, respectively, by
(39)fXx=βabx−a−11+x−a−bβ+1exp−α1+x−ab−1−β1−1+x−a−bβ+1×α1−θ+θexp−α−11+x−ab−1−β1−θ1−exp−α−11+x−ab−1−β2,where x>0,α>0,β>0,0<θ<1,a>0andb>0, and
(40)rx=βabx−a−11+x−a−bβ+1α1−θ+θexp−α−11+x−ab−1−β1−1+x−a−bβ+11−θ1−exp−α−11+x−ab−1−β,x>0.The density plot of the HMWBIII distribution exhibited a variety of shapes such as; reverseJ-shape, J-shape, right skewed, various forms of symmetric, and left skewed shapes as shown in Figure 1. The hazard rate function also showed varied shapes such as; upside-down bathtub, monotone decreasing, and various forms of monotone increasing failure rates for some selected values.Figure 1
Plots of density and hazard rate functions of the HMWBIII distribution.The quantile functionQGu for the HMWBIII distribution is given by
(41)1−u1−θexp−α−11+x−ab−1−β−exp−α1+x−ab−1−β=0,u∈0,1.


### 5.2. Harmonic Mixture Weibull Lomax Distribution

Considering the Lomax distribution as the baseline model with CDF and PDF, respectively, defined by Lomax [21] as Gx=1−1+ax−b and gx=ab1+ax−b+1,x>0,a>0,b>0, the PDF of the Harmonic mixture Weibull Lomax (HMWL) distribution is given by
(42)fXx=βab1+axbβ−11−1+ax−bβ−1exp−α1+axb−1β×α1−θ+θexp−α−11+axb−1β1−θ1−exp−α−11+axb−1β2,x>0,where x>0,α>0,β>0,0<θ<1,a>0andb>0. The hazard function is given by
(43)rx=βab1+axbβ−11−1+ax−bβ−1α1−θ+θexp−α−11+axb−1β1−θ1−exp−α−11+axb−1β,x>0.The density function of HMWL distribution exhibited a wide variety shapes which includeJ-shape, reverse J-shape, right skewed, left skewed, and different forms of symmetric shapes as shown in Figure 2. The hazard rate function also exhibited varied shapes such as, upside down bathtub, monotone decreasing, and different forms of monotone increasing failure rates.Figure 2
Plots of density and hazard rate functions of the HMWL distribution.The quantile functionQGu for the HMWL distribution is given by
(44)1−u1−θexp−α−11+axb−1β−exp−α1+axb−1β=0.


### 5.3. Harmonic Mixture Weibull Weibull Distribution

Consider that the Weibull distribution is a baseline model with CDF and PDF, respectively, defined by Weibull [22] as Gx=1−e−ax−b and gx=abxb−1e−axb,x>0,a>0,b>0. Then, the PDF of the Harmonic mixture Weibull Weibull (HMWW) distribution is given by
(45)fx=βabxb−11−e−axbβ−1expβaxb−αeaxb−1βα1−θ+θexp−α−1eaxb−1β1−θ1−exp−α−1eaxb−1β2,where x>0,α>0,β>0,0<θ<1,a>0andb>0. The hazard function is given by
(46)rx=βabxb−1eβaxb1−e−axbβ−1α1−θ+θexp−α−1eaxb−1β1−θ1−exp−α−1eaxb−1β,x>0.The density plot of the HMWW distribution showed a wide variety of shapes such as; reverseJ-shape, J-shape, right skewed, left skewed, and symmetric (with various levels of kurtosis) as shown in Figure 3. The hazard rate function also showed varying shapes such as, bathtub, upside-down bathtub, monotone decreasing, and various forms of monotone increasing failure rates for some selected parameter values.Figure 3
Plots of density and hazard rate functions of the HMWW distribution.The quantile functionQGu for the Harmonic mixture Weibull Weibull distribution is given by
(47)1−u1−θexp−α−1eaxb−1β−exp−αeaxb−1β=0,u∈0,1.


### 5.4. Harmonic Mixture Weibull Fréchet Distribution

Considering the Fréchet distribution as a baseline model with CDF and PDF, respectively, defined by Fréchet [23] as Gx=e−ax−b and gx=abx−b−1e−ax−b,x>0,a>0,b>0, the PDF of the Harmonic mixture Weibull Fréchet (HMWF) distribution is given by
(48)fx=βabx−b−1exp−βax−b+αeax−b−1−βα1−θ+θexp−α−1eax−b−1−β1−e−ax−bβ+11−θ1−exp−α−1eax−b−1−β2,where x>0,α>0,β>0,0<θ<1,a>0andb>0. The hazard function is given by
(49)rx=βabx−b−1e−βax−bα1−θ+θexp−α−1eax−b−1−β1−e−ax−bβ+11−θ1−exp−α−1eax−b−1−β,x>0.The density plot of the HMWF distribution exhibited a wide variety of attractive shapes such as,J-shape, reverse J-shape, right skewed, left skewed, and various forms of symmetric shapes as shown in Figure 4. The hazard rate function also showed varying shapes such as; upside-down bathtub, monotone decreasing and different forms of monotone increasing failure rates for some selected parameter values.Figure 4
Plots of density and hazard rate functions of the HMWF distribution.The quantile functionQGu for the Harmonic mixture Weibull Fréchet distribution is given by
(50)1−u1−θexp−α−1eax−b−1−β−exp−αeax−b−1−β=0,u∈0,1


### 5.5. Harmonic Mixture Weibull Normal Distribution

Consider that the Normal distribution is a baseline model with the CDF and PDF, respectively, given byGx=Φx−μ/σ and gx=1/σ2πe−1/2x−μ/σ2, −∞<x<∞, −∞<μ<∞, σ>0. Then, the PDF of the Harmonic mixture Weibull Normal (HMWN) distribution is given by
(51)fx=βΦx−μσβ−1exp−12x−μσ2+αΦx−μσ−1−1−β×α1−θ+θexp−α−1Φx−μ/σ−1−1−βσ2π1−Φx−μ/σβ+11−θ1−exp−α−1Φx−μ/σ−1−1−β2,where −∞<x<∞,α>0,β>0,0<θ<1,−∞<μ<∞andσ>0. The hazard function is given by
(52)rx=βe−1/2x−μ/σ2Φx−μ/σβ−1α1−θ+θexp−α−1Φx−μ/σ−1−1−βσ2π1−Φx−μ/σβ+11−θ1−exp−α−1Φx−μ/σ−1−1−β,−∞<x<∞.The density plot of the HMWN distribution exhibited a very wide variety of attractive shapes such as unimodal right skewed, unimodal left skewed, symmetric (with different levels of kurtosis),J-shape, reverse J-shape, bimodal (with different levels of kurtosis), bimodal left skewed, and N-shapes as shown in Figure 5. The hazard rate function also showed a wide variety of very flexible shapes such as, bathtub, upside-down bathtub, various forms of modified upside-down bathtubs, monotone decreasing, and different forms of monotone increasing failure rates for some selected values as shown in Figure 6.Figure 5
Plots of density function of the HMWN distribution.Figure 6
Plots of hazard rate function of the HMWN distribution.The quantile functionQGu for the Harmonic mixture Weibull Normal distribution is given by
(53)1−u1−θexp−α−1Φx−μσ−1−1−β−exp−αΦx−μσ−1−1−β=0,u∈0,1.


## 5.1. Harmonic Mixture Weibull Burr III Distribution

Suppose that the baseline model of the HMW-G family is the Burr III distribution with CDF and PDF, respectively, defined by Burr [20] as Gx=1+x−a−b and gx=abx−a−11+x−a−b−1,x>0,a>0,b>0. Then, the PDF and hazard function of the HMW-Burr III (HMWBIII) distribution are given, respectively, by
(39)fXx=βabx−a−11+x−a−bβ+1exp−α1+x−ab−1−β1−1+x−a−bβ+1×α1−θ+θexp−α−11+x−ab−1−β1−θ1−exp−α−11+x−ab−1−β2,where x>0,α>0,β>0,0<θ<1,a>0andb>0, and
(40)rx=βabx−a−11+x−a−bβ+1α1−θ+θexp−α−11+x−ab−1−β1−1+x−a−bβ+11−θ1−exp−α−11+x−ab−1−β,x>0.The density plot of the HMWBIII distribution exhibited a variety of shapes such as; reverseJ-shape, J-shape, right skewed, various forms of symmetric, and left skewed shapes as shown in Figure 1. The hazard rate function also showed varied shapes such as; upside-down bathtub, monotone decreasing, and various forms of monotone increasing failure rates for some selected values.Figure 1
Plots of density and hazard rate functions of the HMWBIII distribution.The quantile functionQGu for the HMWBIII distribution is given by
(41)1−u1−θexp−α−11+x−ab−1−β−exp−α1+x−ab−1−β=0,u∈0,1.


## 5.2. Harmonic Mixture Weibull Lomax Distribution

Considering the Lomax distribution as the baseline model with CDF and PDF, respectively, defined by Lomax [21] as Gx=1−1+ax−b and gx=ab1+ax−b+1,x>0,a>0,b>0, the PDF of the Harmonic mixture Weibull Lomax (HMWL) distribution is given by
(42)fXx=βab1+axbβ−11−1+ax−bβ−1exp−α1+axb−1β×α1−θ+θexp−α−11+axb−1β1−θ1−exp−α−11+axb−1β2,x>0,where x>0,α>0,β>0,0<θ<1,a>0andb>0. The hazard function is given by
(43)rx=βab1+axbβ−11−1+ax−bβ−1α1−θ+θexp−α−11+axb−1β1−θ1−exp−α−11+axb−1β,x>0.The density function of HMWL distribution exhibited a wide variety shapes which includeJ-shape, reverse J-shape, right skewed, left skewed, and different forms of symmetric shapes as shown in Figure 2. The hazard rate function also exhibited varied shapes such as, upside down bathtub, monotone decreasing, and different forms of monotone increasing failure rates.Figure 2
Plots of density and hazard rate functions of the HMWL distribution.The quantile functionQGu for the HMWL distribution is given by
(44)1−u1−θexp−α−11+axb−1β−exp−α1+axb−1β=0.


## 5.3. Harmonic Mixture Weibull Weibull Distribution

Consider that the Weibull distribution is a baseline model with CDF and PDF, respectively, defined by Weibull [22] as Gx=1−e−ax−b and gx=abxb−1e−axb,x>0,a>0,b>0. Then, the PDF of the Harmonic mixture Weibull Weibull (HMWW) distribution is given by
(45)fx=βabxb−11−e−axbβ−1expβaxb−αeaxb−1βα1−θ+θexp−α−1eaxb−1β1−θ1−exp−α−1eaxb−1β2,where x>0,α>0,β>0,0<θ<1,a>0andb>0. The hazard function is given by
(46)rx=βabxb−1eβaxb1−e−axbβ−1α1−θ+θexp−α−1eaxb−1β1−θ1−exp−α−1eaxb−1β,x>0.The density plot of the HMWW distribution showed a wide variety of shapes such as; reverseJ-shape, J-shape, right skewed, left skewed, and symmetric (with various levels of kurtosis) as shown in Figure 3. The hazard rate function also showed varying shapes such as, bathtub, upside-down bathtub, monotone decreasing, and various forms of monotone increasing failure rates for some selected parameter values.Figure 3
Plots of density and hazard rate functions of the HMWW distribution.The quantile functionQGu for the Harmonic mixture Weibull Weibull distribution is given by
(47)1−u1−θexp−α−1eaxb−1β−exp−αeaxb−1β=0,u∈0,1.


## 5.4. Harmonic Mixture Weibull Fréchet Distribution

Considering the Fréchet distribution as a baseline model with CDF and PDF, respectively, defined by Fréchet [23] as Gx=e−ax−b and gx=abx−b−1e−ax−b,x>0,a>0,b>0, the PDF of the Harmonic mixture Weibull Fréchet (HMWF) distribution is given by
(48)fx=βabx−b−1exp−βax−b+αeax−b−1−βα1−θ+θexp−α−1eax−b−1−β1−e−ax−bβ+11−θ1−exp−α−1eax−b−1−β2,where x>0,α>0,β>0,0<θ<1,a>0andb>0. The hazard function is given by
(49)rx=βabx−b−1e−βax−bα1−θ+θexp−α−1eax−b−1−β1−e−ax−bβ+11−θ1−exp−α−1eax−b−1−β,x>0.The density plot of the HMWF distribution exhibited a wide variety of attractive shapes such as,J-shape, reverse J-shape, right skewed, left skewed, and various forms of symmetric shapes as shown in Figure 4. The hazard rate function also showed varying shapes such as; upside-down bathtub, monotone decreasing and different forms of monotone increasing failure rates for some selected parameter values.Figure 4
Plots of density and hazard rate functions of the HMWF distribution.The quantile functionQGu for the Harmonic mixture Weibull Fréchet distribution is given by
(50)1−u1−θexp−α−1eax−b−1−β−exp−αeax−b−1−β=0,u∈0,1


## 5.5. Harmonic Mixture Weibull Normal Distribution

Consider that the Normal distribution is a baseline model with the CDF and PDF, respectively, given byGx=Φx−μ/σ and gx=1/σ2πe−1/2x−μ/σ2, −∞<x<∞, −∞<μ<∞, σ>0. Then, the PDF of the Harmonic mixture Weibull Normal (HMWN) distribution is given by
(51)fx=βΦx−μσβ−1exp−12x−μσ2+αΦx−μσ−1−1−β×α1−θ+θexp−α−1Φx−μ/σ−1−1−βσ2π1−Φx−μ/σβ+11−θ1−exp−α−1Φx−μ/σ−1−1−β2,where −∞<x<∞,α>0,β>0,0<θ<1,−∞<μ<∞andσ>0. The hazard function is given by
(52)rx=βe−1/2x−μ/σ2Φx−μ/σβ−1α1−θ+θexp−α−1Φx−μ/σ−1−1−βσ2π1−Φx−μ/σβ+11−θ1−exp−α−1Φx−μ/σ−1−1−β,−∞<x<∞.The density plot of the HMWN distribution exhibited a very wide variety of attractive shapes such as unimodal right skewed, unimodal left skewed, symmetric (with different levels of kurtosis),J-shape, reverse J-shape, bimodal (with different levels of kurtosis), bimodal left skewed, and N-shapes as shown in Figure 5. The hazard rate function also showed a wide variety of very flexible shapes such as, bathtub, upside-down bathtub, various forms of modified upside-down bathtubs, monotone decreasing, and different forms of monotone increasing failure rates for some selected values as shown in Figure 6.Figure 5
Plots of density function of the HMWN distribution.Figure 6
Plots of hazard rate function of the HMWN distribution.The quantile functionQGu for the Harmonic mixture Weibull Normal distribution is given by
(53)1−u1−θexp−α−1Φx−μσ−1−1−β−exp−αΦx−μσ−1−1−β=0,u∈0,1.


## 6. Log-HMWW Location-Scale Regression

In this section, the log-HMWW regression model is presented. Suppose the random variableX follows the HMWW distribution, then Y=logX follows the log Harmonic mixture Weibull Weibull (LHMWW) distribution. Let a=e−μ/σandb=1/σ. Following the given reparameterization, the density function of the LHMWW distribution is
(54)fyy=βσexpy−μσ1−exp−expy−μσβ−1expβexpy−μσ−αexpexpy−μσ−1β×α1−θ+θexp−α−1expexpx−μ/σ−1β1−θ1−exp−α−1expexpx−μ/σ−1β2,y∈ℝ,where −∞<μ<∞ is the location parameter, σ>0, α>0, and 0<θ<1 are the scale parameters and β>0 is the shape parameter. The density plot of the LHMWW distribution exhibited varying shapes such as, J-shape, reverse J-shape, right skewed, symmetric, and left skewed shapes as shown in Figure 7. By these properties, the LHMWW distribution is capable of modelling right skewed, symmetric and left skewed dependent variable with covariates in medical studies.Figure 7
Density plot of the LHMWW distribution.The corresponding survival function to (54) is given by
(55)Sy;α,β,σ,μ=exp−αexpexpx−μ/σ−1β1−θ1−exp−α−1expexpx−μ/σ−1β,y∈ℝ.Suppose thatz=y−μ/σ is the standardized random variable, then the PDF is written as
(56)fyy=βσexpz1−exp−expzβ−1expβexpz−αexpexpz−1β×α1−θ+θexp−α−1expexpz−1β1−θ1−exp−α−1expexpz−1β2,y∈ℝ.By using the LHMWW density, we develop the LHMWW location-scale regression model with the following regression structure(57)yj=vjTγ+σzj,i=1,2,⋯,n,where μ=vjTγ is the location parameter which depends on a particular set of covariates, γ=γ1,γ2,⋯,γkT is a k×1 parameter vector for the regression model, vj=vj1,vj2,⋯,vjkT is the set of covariates, and zj is the error term that follows the LHMWW distribution. The unknown parameters of the LHMWW regression model are estimated using the maximum likelihood estimation procedure. The log-likelihood function of the LHMWW regression model is given by
(58)ℓ=nlogβσ+∑j=1nzj+β−1∑j=1n1−exp−expzj+∑j=1nβexpzj−αexpexpzj−1β+∑j=1nlogα1−θ+θexp−α−1expexpzj−1β−2∑j=1nlog1−θ1−exp−α−1expexpzj−1β,where zj=yj−vjTγ/σandn is the number of observations. By maximizing the log-likelihood function in (58), the parameter estimates of the LHMWW regression model are obtained. The adequacy of the regression model is evaluated by using the Cox-Snell residuals [24]. The Cox-Snell residuals of the LHMWW regression model are given by rj=−logSyjα,β,σ,μ,j=1,2,⋯,n, where Syjα,β,σ,μ is defined as in (55). The Cox-Snell residuals plots are expected to follow the standard exponential distribution if the LHMWW regression model gives a good fit to a data set.


## 7. Simulation

In this section, the finite sample properties of the maximum likelihood estimators of the parameters are investigated using Monte Carlo simulations. The Monte Carlo simulations were performed by using the estimators of the HMWBIII distribution. The quantile function of the HMWBIII distribution was used to generate random samples from the HMWBIII distribution. The simulation experiment was replicated 1000 times for each of the sample sizesn=25, 50, 100, 200, 300, and 600 with parameter values a,b,α,β,θ= I: (0.8, 0.3, 0.7, 0.5, 0.1), II: (0.9, 0.2, 0.6, 0.5, 0.2), III: (0.1, 0.3, 0.8, 1.2, 0.3), and IV: (0.9, 0.2, 0.5, 0.9, 0.1). The average estimate (AE), the average bias (AB), the root mean square error (RMSE), and the coverage probability (CP) were used to assess the performance of the estimators of the parameters. Generally, the AE values converge to the actual parameter values as the sample size increases and the RMSE also decreases as the sample size increases. The AB values also converge to zero (0) with increase in the sample size as shown in Table 1 and Table 2. The CP values for most of the estimators are also observed to revolve around the nominal value of 0.975. These characteristics demonstrate that the maximum likelihood method works very effectively in estimating the parameters of the developed family. It also shows that the estimators of the developed family are asymptotically consistent, efficient and unbiased.Table 1
Simulation results for HMWBIII distribution.
ParameternI: (0.8, 0.3, 0.7, 0.5, 0.1)II: (0.9, 0.2, 0.6, 0.5, 0.2)AEABRMSECPAEABRMSECPa251.89831.09832.36410.97502.04181.14182.46430.9750501.42820.62821.29450.94301.29530.39531.13030.92301001.05230.25230.78430.89101.05340.15340.56930.90702000.86510.06510.41360.98700.94420.04420.37150.88703000.82510.02510.36420.89600.92240.02240.29890.91006000.7961−0.00390.21600.97300.91040.02040.21010.9480b250.36440.06440.34120.96100.21320.01320.17590.9040500.31630.01630.12110.95700.20210.00210.07460.94201000.30360.00360.08040.97000.1940-0.00200.04790.95702000.2960−0.00300.05910.97300.1947-0.00170.03360.96803000.2936−0.00260.04550.97100.1952-0.00140.02910.96706000.2966−0.00200.03360.97700.1955-0.00100.02070.9720α251.36570.66574.67040.95001.00160.40163.64280.9530500.85320.15320.50860.96200.68660.08660.52600.90601000.73180.03180.39220.97600.5595-0.08250.39870.90002000.6588−0.03120.27960.98800.5168-0.08020.31150.89503000.6438−0.03040.21590.89800.5202-0.07980.28030.91806000.6540−0.02040.15510.93700.5154-0.06460.25610.9640β250.3969−0.10310.30370.84800.4180-0.08200.24590.9850500.4407−0.05930.28400.87000.4972-0.02830.24770.96301000.52880.02940.29440.97400.52840.02840.22490.91602000.56060.02060.26340.96700.54270.02270.20270.89603000.55790.01790.22270.90600.53700.01700.17580.95506000.53840.01040.16320.92200.51070.01070.11060.9350θ250.17170.07170.00840.95600.1991-0.00090.00780.9200500.18430.08430.00880.94200.22960.02960.00830.98601000.16730.06730.00810.97500.22190.02190.00780.99502000.15070.05070.00740.87300.21400.01400.00750.78703000.13710.03710.00660.73600.21100.01100.00710.80206000.14930.04930.00660.76400.22620.02620.00710.8090Table 2
Simulation results for HMWBIII distribution.
ParameternIII: (0.1, 0.3, 0.8, 1.2, 0.3)IV: (0.9, 0.2, 0.5, 0.9, 0.1)AEABRMSECPAEABRMSECPa252.44641.44645.97330.98203.01652.11656.31160.9680502.08761.08764.15410.96601.87990.97993.24700.93201001.74100.74102.79870.94201.34000.44001.57360.88402001.49600.49602.11370.89001.05390.15390.63160.98603001.30420.30411.19470.86400.96740.06740.45450.84406001.09880.09880.72680.82000.92270.02270.33660.8440b250.49410.19410.92090.98200.27640.07640.87920.9700500.35910.05910.44130.98400.21230.01220.10560.95701000.32490.02490.15220.98500.20220.00220.06760.96202000.33800.02100.12700.98500.1988−0.00120.04990.96603000.33860.02060.10900.98600.20110.00110.04400.96606000.32300.01300.10290.98100.1973−0.00270.03370.9580α258.44057.640535.96910.85602.51742.017413.05670.9860503.62912.829119.35260.90600.85170.35170.98070.95701001.51090.71091.90170.80400.60490.10490.73910.90202001.54350.70351.81810.97700.5649−0.03510.51470.89403001.40910.60911.65820.94700.4603−0.02470.41250.78706001.24970.44971.54670.74100.4902−0.01980.26680.8130β251.0983−0.10171.48140.97400.7450−0.15500.44560.9030501.1163−0.08371.04440.97600.8046−0.09540.40600.91901001.0757−0.78430.54800.97200.8924−0.00760.37390.95802001.2820−0.07180.59400.95800.93290.03290.28830.94103001.1665−0.03350.55490.95100.92760.02760.02650.91506001.20160.00160.45290.90800.91680.02680.20390.8810θ250.2493−0.25070.01200.90400.17060.07060.00780.9770500.2592−0.24080.01920.82700.18870.08870.00830.81801000.2953−0.20470.01240.82300.20550.07550.00860.97202000.2961−0.20390.01220.76800.18690.07290.00800.64403000.2768−0.20320.01220.72300.17860.06860.00840.64306000.2903−0.10970.01200.67700.16890.06820.00750.6150


## 8. Applications of the HMW-G Family

The applications of the special distributions (HMWL and HMWW) of the HMW-G family to real data sets in medical studies are illustrated in this section. To this end, the special distributions of the family are fitted to real data sets and their performances compared to other competing distributions including generalized inverse Weibull (GIW) distribution [25], odd generalized exponential Weibull (OGEW) distribution [26], generalized odd inverse exponential Weibull (GOIEW), and generalized odd inverse exponential Lomax (GOIEL) distributions [27, 28], and exponentiated Lomax (E-Lx) distribution [29]. The total time on test (TTT) plot due to Aarset [30] is used in assessing the applicability of the special distributions to the real data sets. Goodness of fit tests such as Anderson-Darling (AD) test, Cramér-von Mises (CVM) test, and Kolmogorov-Smirnov (K-S) test as well as Akaike information criterion (AIC), corrected AIC (AICc), Bayesian information criterion (BIC), and the log-likelihood are used to assess the performances of the fitted distributions. The P values of the K-S test are provided. A model with the least values of the goodness of fit measures and highest value of the log-likelihood represents the best fitted model for the data set.


### 8.1. First Application

The first data set represents the ordered survival times of blood cancer patients. The data is found in Abouammoh et al. [31]. It can also be found in Amadu [27] and Amadu et al. [28]. The ordered survival times for 40 patients are given in Table 3.Table 3
Ordered survival times of blood cancer patients.
11546180710621251140815781696181516865106312771455157817352557399241165129014781599179941874398311811357122216031815441789102412221369154916051852The TTT plot in Figure8 indicates that the blood cancer data exhibit an increasing failure rate and hence, the HMW-G family is appropriate to fit the data set.Figure 8
TTT plot for the blood cancer data.In Table4, the maximum likelihood parameter estimates of the fitted distributions is presented.Table 4
Parameter estimates of the fitted models on the blood cancer data.
DistributionParameterStandard errorZ valueP valueHMWLa^=9.5631×1012.9142×10−53.2815×106<2.2000×10−16∗∗∗b^=4.5590×10−24.5879×10−49.9371×101<2.2000×10−16∗∗∗α^=4.2570×1026.2717×10−76.7877×108<2.2000×10−16∗∗∗β^=1.4647×1012.2093×10−36.6297×103<2.2000×10−16∗∗∗θ^=8.7947×10−17.2549×10−21.2122×101<2.2000×10−16∗∗∗HMWWa^=1.8287×10−21.2517×10−21.46100.1440b^=4.1880×10−18.0455×10−25.20541.9360×10−7∗∗∗α^=40.97753.0664×10−21336.3478<2.2000×10−16∗∗∗β^=3.33971.54632.15983.0790×10−2∗θ^=9.0063×10−22.2115×10−14.07254.6510×10−5∗∗∗GIWλ^=20.40526.38823.19421.4000×10−3∗θ^=1.18930.118410.04642.2000×10−16∗∗∗b^=69.04721.587443.49702.2000×10−16∗∗∗OGEWα^=2.77761.01292.74226.1020×10−3∗∗β^=7.5292×10−25.4895×10−21.37160.1702γ^=3.7906×10−13.6964×10−210.2548<2.0000×10−16∗∗∗θ^=2.1594×10−15.3592×10−24.02945.5930×10−5∗∗∗GOIEWα^=1.76760.59992.94673.2000×10−3∗β^=42.24690.01432957.53382.2000×10−16∗∗∗γ^=0.00790.00781.01620.3095θ^=0.87820.13526.49668.2140×10−11∗∗∗GOIELα^=6.78751.3537×10−15.0139×1012.2000×10−16∗∗∗β^=5.5062×1026.6430×10−48.2888×1052.2000×10−16∗∗∗γ^=5.2442×10−31.6279×10−33.22131.3000×10−3∗θ^=3.36834.6765×10−17.20265.9070×10−13∗∗∗E-lxα^=2.0759×1025.0510×10−54.1099×1062.2000×10−16∗∗∗λ^=1.0156×10−14.7377×10−22.14370.0321∗θ^=1.23991.3932×10−18.8992.2000×10−16∗∗∗∗means significant at 5% level of significance.Table5 presents the goodness of fit measures of the fitted distributions on the blood cancer data. The results generally show that the HMWW and HMWL distributions provide better fits to the blood cancer data than the other competing models with the HMWW distribution being the overall best fitted model.Table 5
Log-likelihood, goodness-of-fit statistics, and information criteria of the fitted models for the blood cancer data.
ModelℓAICAICcBICADCVMK-SP valueHMWL-302.8800615.75886007588624.20320.60720.06270.08900.90950HMWW-301.9300613.8636598.8636622.30800.47280.04900.08590.9292GIW-352.3200710.6453711.2607715.92894.23620.76500.26070.0058OGEW-305.7900619.5847606.2514626.34021.00670.12820.13200.4891GOIEW-330.4800668.9273669.9800675.97211.50160.25530.15830.2687GOIEL-338.7500685.5067686.5593692.55150.85180.10920.24770.0102E-lx-351.9400709.8701710.4855715.15374.22100.76210.25200.0085∗bolded means best based on selection criteria.Figure9 shows the densities and CDFs plots of the fitted models. The results give a confirmation that the HMWW distribution provides a better fit to the data than the other competing models.Figure 9
Fitted densities and CDFs plots of the blood cancer data.


### 8.2. Second Application

The second data set represents the survival times (life lengths in years) until onset of diabetes from a random sample of 105 patients obtained from the Bolgatanga Regional Hospital in the Upper East region of Ghana. The data set is shown in Table6.Table 6
Survival times of diabetes patients.
521869192874252956397626813334383834354345456347464242424146454543414049494853533554615455552573517437565858585750186262816347646465676036681969616161707583526233802676753729395135595082525271517324514848405436The TTT plot in Figure10 indicates that the diabetes data exhibit an increasing failure rate and hence, the HMW-G family is appropriate to fit the data set.Figure 10
TTT plot for the diabetes data.In Table7, the maximum likelihood estimates of the parameters of the fitted distributions are presented.Table 7
Parameter estimates of the fitted models on the diabetes data.
DistributionParameter estimateStandard errorZ valueP valueHMWLa^=3.7238×10−31.3636×10−32.73086.3170×10−3∗∗b^=2.24007.0116×10−13.19471.4000×10−3∗∗α^=7.38822.7030×10−2273.3389<2.2000×10−16∗∗∗β^=3.13151.5959×10−119.6221<2.2000×10−16∗∗∗θ^=8.5928×10−21.6698×10−10.51460.6068HMWWa^=3.2731×10−13.5914×10−29.1137<2.2000×10−16∗∗∗b^=1.3193×10−12.3472×10−25.62051.9050×10−8∗∗∗α^=4.3077×1022.0874×10−52.0637×107<2.2000×10−16∗∗∗β^=2.0680×1011.0524×10−21.9651×103<2.2000×10−16∗∗∗θ^=3.3286×10−37.2963×10−14.6000×10−30.9964GIWλ^=14.39497.8816×10−118.2640<2.2000×10−16∗∗∗θ^=2.48801.6693×10−114.9040<2.2000×10−16∗∗∗b^=12.47893.6543×10−134.1490<2.2000×10−16∗∗∗OGEWα^=6.20071.0197×10−2608.0858<2.0000×10−16∗∗∗β^=6.25723.4785×10−31798.8481<2.0000×10−16∗∗∗γ^=1.17859.7083×10−212.1387<2.0000×10−16∗∗∗θ^=3.1583×10−31.2296×10−32.56861.0210×10−2∗GOIEWα^=3.82649.3022×10−14.11343.8990×10−5∗∗∗β^=45.76235.6602×10−2808.4855<2.2000×10−16∗∗∗γ^=2.7834×10−21.6309×10−21.70670.0879θ^=1.26181.4195×10−18.8889<2.2000×10−16∗∗∗GOIELα^=1.1535×1011.32948.6765<2.2000×10−16∗∗∗β^=9.9124×1017.1690×10−21382.6765<2.2000×10−16∗∗∗γ^=1.1493×10−26.3054×10−418.2274<2.2000×10−16∗∗∗θ^=1.0812×1015.0939×10−121.2258<2.2000×10−16∗∗∗E-lxα^=1.7112×1014.3906×10−83.8975×108<2.2000×10−16∗∗∗λ^=7.1402×10−32.8425×10−42.5119×101<2.2000×10−16∗∗∗θ^=1.1041×1012.1284×10−75.1872×107<2.2000×10−16∗∗∗∗means significant at 5% level of significance.Table8 presents the goodness of fit measures of the fitted models. The results show that the special distributions (HMWW and HMWL) of the HMW-G family generally provide better fits to the diabetes data than the other competing models with the HMWW distribution being the overall best fitted model.Table 8
Log-likelihood, goodness-of-fit statistics, and information criteria for the Diabetes data.
ModelℓAICAICcBICADCVMK-SP valueHMWL-438.8100887.6210872.6210900.89080.22640.02830.04310.9899HMWW-438.7500887.5028872.5028900.77260.20750.02220.04240.9916GIW-464.7700935.5496923.5496943.51144.38920.68700.14760.0207OGEW-441.7500891.5075878.1742902.12340.67420.10440.07280.6342GOIEW-442.3900892.7726879.4393903.38851.21950.18850.11110.1500GOIEL-447.9600903.9159890.5826914.53173.16520.57740.13920.0342E-lx-449.8500905.6986893.6986913.66051.95250.28910.10280.2174∗bolded means best based on selection criteria.Figure11 shows the densities and CDFs plots of the fitted models. The results confirm that the HMWW and HMWL distributions of the HMW-G family generally provide better fits to the data than the other competing models.Figure 11
Plots of fitted densities and CDFs of the diabetes data.


### 8.3. Third Application

The third data set represents the survival times (life lengths in years) until onset of hypertension from a random sample of 119 patients obtained from the Bolgatanga Regional Hospital in the Upper East region of Ghana. The data set is shown in Table9.Table 9
Survival times of hypertension patients.
71539625271385635693471667052373571731974747551764919767876764947484846464641404345474744454642434220282660272429602560693669696868676767523566556661616464656563636239626262595959585858185757565637535353535454661750755138526645255195873The TTT plot in Figure12 indicates that the hypertension data exhibit an increasing failure rate and therefore, the HMW-G family is appropriate to fit the data set.Figure 12
TTT plot for the hypertension data.Table10 shows the maximum likelihood estimates of the parameters of the fitted distributions.Table 10
Parameter estimates of the fitted models on the hypertension data.
DistributionParameter estimateStandard errorZ-valueP valueHMWLa^=3.0825×10−32.4265×10−412.7035<2.2000×10−16∗∗∗b^=4.36751.0806×10−140.4177<2.2000×10−16∗∗∗α^=1.0912×10−11.5700×10−10.69500.4871β^=2.94173.0333×10−19.6981<2.2000×10−16∗∗∗θ^=6.2795×10−12.0319×10−13.09051.9980×10−3∗∗HMWWa^=1.2373×10−14.9012×10−22.52440.0116∗b^=2.9675×10−16.8779×10−24.31461.5990×10−5∗∗∗α^=2.0787×1023.6742×10−356575.9248<2.2000×10−16∗∗∗β^=6.47441.83463.52904.1710×10−4∗∗∗θ^=8.9463×10−19.1076×10−29.8230<2.2000×10−16∗∗∗GIWλ^=3.9662×10−19.0450×10−24.38501.1600×10−5∗∗∗θ^=1.30757.0211×10−21.8622×101<2.2000×10−16∗∗∗b^=3.7129×1027.3899×10−55.0244×106<2.2000×10−16∗∗∗OGEWα^=2.15881.1590×10−118.6271<2.0000×10−16∗∗∗β^=6.5927×10−12.5204×10−12.61578.9030×10−3∗∗γ^=1.57251.7709×10−18.8798<2.0000×10−16∗∗∗θ^=2.0899×10−31.9202×10−31.08840.2764GOIEWα^=6.0373×10−11.1470×10−15.26341.4140×10−7∗∗∗β^=13.60551.1507×10−21182.3470<2.2000×10−16∗∗∗γ^=4.5046×10−33.6406×10−31.23730.2160θ^=1.62971.9507×10−18.3542<2.2000×10−16∗∗∗GOIELα^=1.90422.6522×10−17.17966.9910×10−13∗∗∗β^=9.8331×1019.2359×10−41.0647×105<2.2000×10−16∗∗∗γ^=8.7185×10−32.6638×10−43.2729×101<2.2000×10−16∗∗∗θ^=1.3273×1015.1568×10−32.5739×103<2.2000×10−16∗∗∗E-lxα^=7.06541.14556.16806.9150×10−10∗∗∗λ^=3.8023×10−32.9418×10−412.9250<2.2000×10−16∗∗∗θ^=1.4008×108.3986×10−2166.7850<2.2000×10−16∗∗∗∗means significant at 5% significance level.The goodness of fit measures of the fitted models on the hypertension data are given in Table11. The results show that the HMWW and HMWL distributions of the HMW-G family generally provide better fits to the data than the other competing models with the HMWW distribution being the overall best fitted model.Table 11
Log-likelihood, goodness-of-fit statistics, and information criteria of the fitted models for hypertension data.
ModelℓAICAICcBICADCVMK-SP valueHMWL−497.97001005.9490990.94901019.84500.81760.04850.05360.8834HMWW-496.42001002.8500987.85001016.74600.62990.05750.05090.9170GIW−592.61001191.22001179.22001199.557015.91102.96050.31111.974×10−10OGEW−499.69001007.3730994.03971018.49001.11560.12110.06560.6847GOIEW−513.02001034.03401020.70101045.15108.14301.53630.19660.0002GOIEL−531.98001071.96001058.62701083.077014.00202.77260.24311.5620×10−6E-lx−538.17001078.33401066.33401086.67107.34461.29760.17670.0012∗bolded means best based on selection criteria.The densities and CDFs plots of the fitted models are shown in Figure13. The results give a confirmation that the HMWW distribution provides a better fit to the data than the other competing models.Figure 13
Plots of fitted densities and CDFs of the hypertension data.


### 8.4. Fourth Application

In this section, the application of the LHMWW location-scale regression model is demonstrated by modelling a real data set. The data set obtained from the Bolgatanga Regional Hospital in the Upper East Region of Ghana represents the survival times (life lengths in years) until onset of hypertension from a random sample of 119 patients with gender as a covariate. The gender (1=male, 0=female) is presented in brackets for each survival time. The data set is given in Table 12.Table 12
Hypertension data with covariate (gender).
71(1)5(1)39(1)62(1)52(0)71(0)38(0)56(1)35(1)69(1)34(1)71(1)66(0)70(1)52(0)37(0)35(0)71(1)73(1)19(0)74(0)74(1)75(1)51(0)76(1)49(0)19(1)76(0)78(1)76(0)76(0)49(1)47(1)48(0)48(0)46(0)46(1)46(1)41(0)40(0)43(1)45(0)47(0)47(0)44(0)45(1)46(1)42(1)43(0)42(0)20(1)28(0)26(0)60(0)27(1)24(0)29(0)60(1)25(1)60(1)69(1)36(1)69(0)69(1)68(0)68(0)67(1)67(0)67(0)52(0)35(0)66(0)55(0)66(1)61(1)61(0)64(0)64(0)65(0)65(0)63(1)63(1)62(0)39(1)62(0)62(0)62(0)59(1)59(0)59(1)58(0)58(0)58(0)18(1)57(0)57(0)56(0)56(0)37(1)53(0)53(0)53(0)53(1)54(1)54(1)66(0)17(0)50(0)75(0)51(0)38(0)52(1)66(0)4(1)52(0)55(0)19(1)58(1)73(0)The dependent variable, time until the onset of hypertensionyj, is modelled with gender xj1 (1=male, 0=female) as the covariate. To this end, the following regression model is fitted to the data set
(59)yj=γ0+γ1vj1+σzj,where yj follows the LHMWW distribution. The performance of the LHMWW regression model was assessed by comparing with the log Marshal-Olkin Weibull Weibull (LMOWW) regression model. The parameter estimates of the regression models are presented in Table 13. The goodness of fit measures of the regression models show that the LHMWW regression model performs better than the LMOWW regression model. From the parameter estimates of the LHMWW regression model, gender is statistically significant at the 5% level of significance. Thus, the LHMWWW regression results show that the time frame for onset of hypertension is not the same in males and females, and this is evidenced by the significant influence of gender on the survival times of hypertension. This is a revelation of gender differences in relation to time until the onset of hypertension. The finding is useful and consistent with that of Paresh et al. [32].Table 13
Parameter estimates of the regression models for the hypertension data.
ModelParameterEstimateStandard errorZ valueP valueLHMWWσ4.24041.07903.92988.5010×10−5α345.27332.8976×10−11191.6015<2.2000×10−16β3.2153×10−19.2454×10−23.47775.0570×10−4θ2.6481×10−27.2045×10−10.03680.9707γ0136.66818.128516.8134<2.2000×10−16γ14.1023×10−17.1629×10−25.72711.0220×10−8ℓ= -493.5100AIC=999.0204BIC=1015.6950K−S=0.04870.9407LMOWWσ3.5779×1021.6212×10−222068.5615<2.2000×10−16β7.14052.5708×10−127.7749<2.2000×10−16θ2.1458×10−25.3641×10−34.00026.3280×10−5γ01.3834×1025.1886×10−22666.1188<2.2000×10−16γ19.8022×10−12.59660.37750.7058ℓ=497.4500AIC=1004.9000BIC=1018.7950K−S=0.9644<2.2000×10−16The likelihood ratio test (LRT) was also performed to compare the LHMWW regression model and the LMOWW regression model. The LRT statistic of 7.8791 with aP value of 0.0050 showed that the LHMWW regression model performs better than the LMOWW regression model.The Cox-Snell residuals were used to assess the adequacy of the LHMWW regression model. The P-P plot results in Figure14 show that the LHMWW regression model provides a very good fit to the data set and therefore can be adequately applied for modelling real life data in medical studies.Figure 14
P-P plot of the Cox-Snell Residuals for LHMWW regression residuals.


## 8.1. First Application

The first data set represents the ordered survival times of blood cancer patients. The data is found in Abouammoh et al. [31]. It can also be found in Amadu [27] and Amadu et al. [28]. The ordered survival times for 40 patients are given in Table 3.Table 3
Ordered survival times of blood cancer patients.
11546180710621251140815781696181516865106312771455157817352557399241165129014781599179941874398311811357122216031815441789102412221369154916051852The TTT plot in Figure8 indicates that the blood cancer data exhibit an increasing failure rate and hence, the HMW-G family is appropriate to fit the data set.Figure 8
TTT plot for the blood cancer data.In Table4, the maximum likelihood parameter estimates of the fitted distributions is presented.Table 4
Parameter estimates of the fitted models on the blood cancer data.
DistributionParameterStandard errorZ valueP valueHMWLa^=9.5631×1012.9142×10−53.2815×106<2.2000×10−16∗∗∗b^=4.5590×10−24.5879×10−49.9371×101<2.2000×10−16∗∗∗α^=4.2570×1026.2717×10−76.7877×108<2.2000×10−16∗∗∗β^=1.4647×1012.2093×10−36.6297×103<2.2000×10−16∗∗∗θ^=8.7947×10−17.2549×10−21.2122×101<2.2000×10−16∗∗∗HMWWa^=1.8287×10−21.2517×10−21.46100.1440b^=4.1880×10−18.0455×10−25.20541.9360×10−7∗∗∗α^=40.97753.0664×10−21336.3478<2.2000×10−16∗∗∗β^=3.33971.54632.15983.0790×10−2∗θ^=9.0063×10−22.2115×10−14.07254.6510×10−5∗∗∗GIWλ^=20.40526.38823.19421.4000×10−3∗θ^=1.18930.118410.04642.2000×10−16∗∗∗b^=69.04721.587443.49702.2000×10−16∗∗∗OGEWα^=2.77761.01292.74226.1020×10−3∗∗β^=7.5292×10−25.4895×10−21.37160.1702γ^=3.7906×10−13.6964×10−210.2548<2.0000×10−16∗∗∗θ^=2.1594×10−15.3592×10−24.02945.5930×10−5∗∗∗GOIEWα^=1.76760.59992.94673.2000×10−3∗β^=42.24690.01432957.53382.2000×10−16∗∗∗γ^=0.00790.00781.01620.3095θ^=0.87820.13526.49668.2140×10−11∗∗∗GOIELα^=6.78751.3537×10−15.0139×1012.2000×10−16∗∗∗β^=5.5062×1026.6430×10−48.2888×1052.2000×10−16∗∗∗γ^=5.2442×10−31.6279×10−33.22131.3000×10−3∗θ^=3.36834.6765×10−17.20265.9070×10−13∗∗∗E-lxα^=2.0759×1025.0510×10−54.1099×1062.2000×10−16∗∗∗λ^=1.0156×10−14.7377×10−22.14370.0321∗θ^=1.23991.3932×10−18.8992.2000×10−16∗∗∗∗means significant at 5% level of significance.Table5 presents the goodness of fit measures of the fitted distributions on the blood cancer data. The results generally show that the HMWW and HMWL distributions provide better fits to the blood cancer data than the other competing models with the HMWW distribution being the overall best fitted model.Table 5
Log-likelihood, goodness-of-fit statistics, and information criteria of the fitted models for the blood cancer data.
ModelℓAICAICcBICADCVMK-SP valueHMWL-302.8800615.75886007588624.20320.60720.06270.08900.90950HMWW-301.9300613.8636598.8636622.30800.47280.04900.08590.9292GIW-352.3200710.6453711.2607715.92894.23620.76500.26070.0058OGEW-305.7900619.5847606.2514626.34021.00670.12820.13200.4891GOIEW-330.4800668.9273669.9800675.97211.50160.25530.15830.2687GOIEL-338.7500685.5067686.5593692.55150.85180.10920.24770.0102E-lx-351.9400709.8701710.4855715.15374.22100.76210.25200.0085∗bolded means best based on selection criteria.Figure9 shows the densities and CDFs plots of the fitted models. The results give a confirmation that the HMWW distribution provides a better fit to the data than the other competing models.Figure 9
Fitted densities and CDFs plots of the blood cancer data.


## 8.2. Second Application

The second data set represents the survival times (life lengths in years) until onset of diabetes from a random sample of 105 patients obtained from the Bolgatanga Regional Hospital in the Upper East region of Ghana. The data set is shown in Table6.Table 6
Survival times of diabetes patients.
521869192874252956397626813334383834354345456347464242424146454543414049494853533554615455552573517437565858585750186262816347646465676036681969616161707583526233802676753729395135595082525271517324514848405436The TTT plot in Figure10 indicates that the diabetes data exhibit an increasing failure rate and hence, the HMW-G family is appropriate to fit the data set.Figure 10
TTT plot for the diabetes data.In Table7, the maximum likelihood estimates of the parameters of the fitted distributions are presented.Table 7
Parameter estimates of the fitted models on the diabetes data.
DistributionParameter estimateStandard errorZ valueP valueHMWLa^=3.7238×10−31.3636×10−32.73086.3170×10−3∗∗b^=2.24007.0116×10−13.19471.4000×10−3∗∗α^=7.38822.7030×10−2273.3389<2.2000×10−16∗∗∗β^=3.13151.5959×10−119.6221<2.2000×10−16∗∗∗θ^=8.5928×10−21.6698×10−10.51460.6068HMWWa^=3.2731×10−13.5914×10−29.1137<2.2000×10−16∗∗∗b^=1.3193×10−12.3472×10−25.62051.9050×10−8∗∗∗α^=4.3077×1022.0874×10−52.0637×107<2.2000×10−16∗∗∗β^=2.0680×1011.0524×10−21.9651×103<2.2000×10−16∗∗∗θ^=3.3286×10−37.2963×10−14.6000×10−30.9964GIWλ^=14.39497.8816×10−118.2640<2.2000×10−16∗∗∗θ^=2.48801.6693×10−114.9040<2.2000×10−16∗∗∗b^=12.47893.6543×10−134.1490<2.2000×10−16∗∗∗OGEWα^=6.20071.0197×10−2608.0858<2.0000×10−16∗∗∗β^=6.25723.4785×10−31798.8481<2.0000×10−16∗∗∗γ^=1.17859.7083×10−212.1387<2.0000×10−16∗∗∗θ^=3.1583×10−31.2296×10−32.56861.0210×10−2∗GOIEWα^=3.82649.3022×10−14.11343.8990×10−5∗∗∗β^=45.76235.6602×10−2808.4855<2.2000×10−16∗∗∗γ^=2.7834×10−21.6309×10−21.70670.0879θ^=1.26181.4195×10−18.8889<2.2000×10−16∗∗∗GOIELα^=1.1535×1011.32948.6765<2.2000×10−16∗∗∗β^=9.9124×1017.1690×10−21382.6765<2.2000×10−16∗∗∗γ^=1.1493×10−26.3054×10−418.2274<2.2000×10−16∗∗∗θ^=1.0812×1015.0939×10−121.2258<2.2000×10−16∗∗∗E-lxα^=1.7112×1014.3906×10−83.8975×108<2.2000×10−16∗∗∗λ^=7.1402×10−32.8425×10−42.5119×101<2.2000×10−16∗∗∗θ^=1.1041×1012.1284×10−75.1872×107<2.2000×10−16∗∗∗∗means significant at 5% level of significance.Table8 presents the goodness of fit measures of the fitted models. The results show that the special distributions (HMWW and HMWL) of the HMW-G family generally provide better fits to the diabetes data than the other competing models with the HMWW distribution being the overall best fitted model.Table 8
Log-likelihood, goodness-of-fit statistics, and information criteria for the Diabetes data.
ModelℓAICAICcBICADCVMK-SP valueHMWL-438.8100887.6210872.6210900.89080.22640.02830.04310.9899HMWW-438.7500887.5028872.5028900.77260.20750.02220.04240.9916GIW-464.7700935.5496923.5496943.51144.38920.68700.14760.0207OGEW-441.7500891.5075878.1742902.12340.67420.10440.07280.6342GOIEW-442.3900892.7726879.4393903.38851.21950.18850.11110.1500GOIEL-447.9600903.9159890.5826914.53173.16520.57740.13920.0342E-lx-449.8500905.6986893.6986913.66051.95250.28910.10280.2174∗bolded means best based on selection criteria.Figure11 shows the densities and CDFs plots of the fitted models. The results confirm that the HMWW and HMWL distributions of the HMW-G family generally provide better fits to the data than the other competing models.Figure 11
Plots of fitted densities and CDFs of the diabetes data.


## 8.3. Third Application

The third data set represents the survival times (life lengths in years) until onset of hypertension from a random sample of 119 patients obtained from the Bolgatanga Regional Hospital in the Upper East region of Ghana. The data set is shown in Table9.Table 9
Survival times of hypertension patients.
71539625271385635693471667052373571731974747551764919767876764947484846464641404345474744454642434220282660272429602560693669696868676767523566556661616464656563636239626262595959585858185757565637535353535454661750755138526645255195873The TTT plot in Figure12 indicates that the hypertension data exhibit an increasing failure rate and therefore, the HMW-G family is appropriate to fit the data set.Figure 12
TTT plot for the hypertension data.Table10 shows the maximum likelihood estimates of the parameters of the fitted distributions.Table 10
Parameter estimates of the fitted models on the hypertension data.
DistributionParameter estimateStandard errorZ-valueP valueHMWLa^=3.0825×10−32.4265×10−412.7035<2.2000×10−16∗∗∗b^=4.36751.0806×10−140.4177<2.2000×10−16∗∗∗α^=1.0912×10−11.5700×10−10.69500.4871β^=2.94173.0333×10−19.6981<2.2000×10−16∗∗∗θ^=6.2795×10−12.0319×10−13.09051.9980×10−3∗∗HMWWa^=1.2373×10−14.9012×10−22.52440.0116∗b^=2.9675×10−16.8779×10−24.31461.5990×10−5∗∗∗α^=2.0787×1023.6742×10−356575.9248<2.2000×10−16∗∗∗β^=6.47441.83463.52904.1710×10−4∗∗∗θ^=8.9463×10−19.1076×10−29.8230<2.2000×10−16∗∗∗GIWλ^=3.9662×10−19.0450×10−24.38501.1600×10−5∗∗∗θ^=1.30757.0211×10−21.8622×101<2.2000×10−16∗∗∗b^=3.7129×1027.3899×10−55.0244×106<2.2000×10−16∗∗∗OGEWα^=2.15881.1590×10−118.6271<2.0000×10−16∗∗∗β^=6.5927×10−12.5204×10−12.61578.9030×10−3∗∗γ^=1.57251.7709×10−18.8798<2.0000×10−16∗∗∗θ^=2.0899×10−31.9202×10−31.08840.2764GOIEWα^=6.0373×10−11.1470×10−15.26341.4140×10−7∗∗∗β^=13.60551.1507×10−21182.3470<2.2000×10−16∗∗∗γ^=4.5046×10−33.6406×10−31.23730.2160θ^=1.62971.9507×10−18.3542<2.2000×10−16∗∗∗GOIELα^=1.90422.6522×10−17.17966.9910×10−13∗∗∗β^=9.8331×1019.2359×10−41.0647×105<2.2000×10−16∗∗∗γ^=8.7185×10−32.6638×10−43.2729×101<2.2000×10−16∗∗∗θ^=1.3273×1015.1568×10−32.5739×103<2.2000×10−16∗∗∗E-lxα^=7.06541.14556.16806.9150×10−10∗∗∗λ^=3.8023×10−32.9418×10−412.9250<2.2000×10−16∗∗∗θ^=1.4008×108.3986×10−2166.7850<2.2000×10−16∗∗∗∗means significant at 5% significance level.The goodness of fit measures of the fitted models on the hypertension data are given in Table11. The results show that the HMWW and HMWL distributions of the HMW-G family generally provide better fits to the data than the other competing models with the HMWW distribution being the overall best fitted model.Table 11
Log-likelihood, goodness-of-fit statistics, and information criteria of the fitted models for hypertension data.
ModelℓAICAICcBICADCVMK-SP valueHMWL−497.97001005.9490990.94901019.84500.81760.04850.05360.8834HMWW-496.42001002.8500987.85001016.74600.62990.05750.05090.9170GIW−592.61001191.22001179.22001199.557015.91102.96050.31111.974×10−10OGEW−499.69001007.3730994.03971018.49001.11560.12110.06560.6847GOIEW−513.02001034.03401020.70101045.15108.14301.53630.19660.0002GOIEL−531.98001071.96001058.62701083.077014.00202.77260.24311.5620×10−6E-lx−538.17001078.33401066.33401086.67107.34461.29760.17670.0012∗bolded means best based on selection criteria.The densities and CDFs plots of the fitted models are shown in Figure13. The results give a confirmation that the HMWW distribution provides a better fit to the data than the other competing models.Figure 13
Plots of fitted densities and CDFs of the hypertension data.


## 8.4. Fourth Application

In this section, the application of the LHMWW location-scale regression model is demonstrated by modelling a real data set. The data set obtained from the Bolgatanga Regional Hospital in the Upper East Region of Ghana represents the survival times (life lengths in years) until onset of hypertension from a random sample of 119 patients with gender as a covariate. The gender (1=male, 0=female) is presented in brackets for each survival time. The data set is given in Table 12.Table 12
Hypertension data with covariate (gender).
71(1)5(1)39(1)62(1)52(0)71(0)38(0)56(1)35(1)69(1)34(1)71(1)66(0)70(1)52(0)37(0)35(0)71(1)73(1)19(0)74(0)74(1)75(1)51(0)76(1)49(0)19(1)76(0)78(1)76(0)76(0)49(1)47(1)48(0)48(0)46(0)46(1)46(1)41(0)40(0)43(1)45(0)47(0)47(0)44(0)45(1)46(1)42(1)43(0)42(0)20(1)28(0)26(0)60(0)27(1)24(0)29(0)60(1)25(1)60(1)69(1)36(1)69(0)69(1)68(0)68(0)67(1)67(0)67(0)52(0)35(0)66(0)55(0)66(1)61(1)61(0)64(0)64(0)65(0)65(0)63(1)63(1)62(0)39(1)62(0)62(0)62(0)59(1)59(0)59(1)58(0)58(0)58(0)18(1)57(0)57(0)56(0)56(0)37(1)53(0)53(0)53(0)53(1)54(1)54(1)66(0)17(0)50(0)75(0)51(0)38(0)52(1)66(0)4(1)52(0)55(0)19(1)58(1)73(0)The dependent variable, time until the onset of hypertensionyj, is modelled with gender xj1 (1=male, 0=female) as the covariate. To this end, the following regression model is fitted to the data set
(59)yj=γ0+γ1vj1+σzj,where yj follows the LHMWW distribution. The performance of the LHMWW regression model was assessed by comparing with the log Marshal-Olkin Weibull Weibull (LMOWW) regression model. The parameter estimates of the regression models are presented in Table 13. The goodness of fit measures of the regression models show that the LHMWW regression model performs better than the LMOWW regression model. From the parameter estimates of the LHMWW regression model, gender is statistically significant at the 5% level of significance. Thus, the LHMWWW regression results show that the time frame for onset of hypertension is not the same in males and females, and this is evidenced by the significant influence of gender on the survival times of hypertension. This is a revelation of gender differences in relation to time until the onset of hypertension. The finding is useful and consistent with that of Paresh et al. [32].Table 13
Parameter estimates of the regression models for the hypertension data.
ModelParameterEstimateStandard errorZ valueP valueLHMWWσ4.24041.07903.92988.5010×10−5α345.27332.8976×10−11191.6015<2.2000×10−16β3.2153×10−19.2454×10−23.47775.0570×10−4θ2.6481×10−27.2045×10−10.03680.9707γ0136.66818.128516.8134<2.2000×10−16γ14.1023×10−17.1629×10−25.72711.0220×10−8ℓ= -493.5100AIC=999.0204BIC=1015.6950K−S=0.04870.9407LMOWWσ3.5779×1021.6212×10−222068.5615<2.2000×10−16β7.14052.5708×10−127.7749<2.2000×10−16θ2.1458×10−25.3641×10−34.00026.3280×10−5γ01.3834×1025.1886×10−22666.1188<2.2000×10−16γ19.8022×10−12.59660.37750.7058ℓ=497.4500AIC=1004.9000BIC=1018.7950K−S=0.9644<2.2000×10−16The likelihood ratio test (LRT) was also performed to compare the LHMWW regression model and the LMOWW regression model. The LRT statistic of 7.8791 with aP value of 0.0050 showed that the LHMWW regression model performs better than the LMOWW regression model.The Cox-Snell residuals were used to assess the adequacy of the LHMWW regression model. The P-P plot results in Figure14 show that the LHMWW regression model provides a very good fit to the data set and therefore can be adequately applied for modelling real life data in medical studies.Figure 14
P-P plot of the Cox-Snell Residuals for LHMWW regression residuals.


## 9. Conclusion

A new family of probability distributions called the Harmonic mixture Weibull-G (HMW-G) family of distributions is introduced in this work. The statistical properties of the family including quantile function, moments, incomplete moment, moment generating function, and mean residual life were comprehensively derived. Five special distributions (HMWBIII, HMWL, HMWW, HMWF, and HMWN) of the family were developed and studied. The density plots of the special distributions showed a wide variety of very attractive shapes making them very suitable for modelling bimodal data sets as well as left skewed, right skewed, and symmetric data sets in medical studies. The hazard function plots also showed a wide variety of shapes making the family very suitable for modelling data with both monotone and nonmonotone failure rates. The maximum likelihood method was used in estimating the parameters of the HMW-G family. The performance of the maximum likelihood estimators was assessed using Monte-Carlo simulation studies. The LHMWW location-scale regression model was developed to investigate the effect of covariates on a response variable that follows the LHMWW distribution. The usefulness of the HMW-G family was demonstrated with applications to real data sets in medicine. The applications empirically showed that the HMWW distribution provides a better fit to the given data sets than the other competing models. Finally, the application of the regression model showed that the LHMWW regression model provided a very good fit to the given data and hence can be adequately applied for modelling data in medical studies. As part of our future studies, sensitivity analysis of the regression model will be performed.



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