Abstract
The main idea of this paper is to present families of bivariate distributions that depend in their formation on adding a shape parameter to the powers of the hazard and reversed hazard functions in different manners, which would provide additional flexibility in applications. Different baseline distributions were used namely, exponential, inverse exponential, uniform, inverse uniform, inverse Rayleigh, Gompertz and Pareto. Many of the mathematical properties of these families are discussed in detail. Moreover, it is observed that the new bivariate distributions also can make appropriate modeling of three real data sets.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The hazard and reversed hazard rates play important roles in the statistical literatures because of their applicability in many fields. The concept of hazard rate is very well known in the literature and lifetime distributions are usually characterized using the concept of failure rate h(t), defined as
which can be equivalently written as
where f(t)and F(t) are the pdf and cdf of life time T.
The failure rate h(t) measures the instantaneous rate of failure or death at time t, given that an individual survives up to time t. The failure rate is also known as conditional failure rate in reliability, the hazard rate in survival analysis, the force of mortality in demography, the age-specific failure rate in epidemiology. In extreme-value theory, it is known as the intensity rate and its reciprocal is termed as Mill's ratio in economics.
It can be shown that h(x) uniquely determines the distribution. When X is non-negative and has a distribution function absolutely continuous with respect to the Lebesgue measure h(t) can provides
Where H(x) is the cumulative hazard rate.
In many practical situations reversed hazard (RH) rate is more appropriate to analyze the survival data. Reversed hazard rate was proposed as a dual to the hazard rate respectively, as
The reversed hazard rate specifies the instantaneous rate of death or failure at time t, given that it failed before time t. Thus in a small interval, r(t)∆t is the approximate probability of failure in the interval (t − ∆t, t], given failure before the end of the interval.
It can be shown that r(t) determines the cdf through the following relation
Where R(x) = logF(x) denotes the cumulative reversed hazard rate.
There are many methods for adding a shape parameter to a family of distributions based on the survival and failure functions that produced the so-called, proportional hazard family and proportional reversed hazard family, along the same line the hazard and reversed hazard functions can also be used to adding a power parameter (shape parameter) that are producing two important families of distributions namely, hazard power parameter and reversed hazard power parameter. The aim of this paper is to introduce the bivariate extensions of these families based on an idea similar to that of Theorem 3.2 proposed by Marshall and Olkin (1967). These authors introduced a multivariate exponential distribution whose marginals have exponential distributions and proposed a bivariate Weibull distribution. The proposed bivariate distributions are constructed from three independent distributions using both minimization and maximization process. These new distributions are singular distributions, and they can be used quit conveniently if there are ties in the data.
Our new models can be extended to different data sets for example, survival times of failure of paired organs like kidneys, lungs, eyes, ears, dental implants etc. In industrial applications, the breakdown times of dual generators in a power plant or failure times of two engines in a two-engine airplanes. In the infectious diseases, time to infection of two or more family members who might visit an infected person and all of them become infected. Some examples are the human lifetimes for which natural disasters or accidents lead to the death of several persons at the same time. They are also widely used in life insurance and the design of multiple life insurance products. Furthermore, they are used in statistics and reliability in shock model, competing risks model, stress model, maintenance model and longevity model, as well as warranty polices based on failure time and warranty servicing time.
The paper is organized as follows: In Section 2, some baseline distributions are introduced. The bivariate reversed hazard power parameter (BRPP) family of distributions is introduced in Section 3. The bivariate hazard power parameter (BHPP) family of distributions is introduced in Section 4. The bivariate power parameter (BPP) family of distributions is introduced Section 5. The bivariate proportional hazard (BPHP) family of distributions is discussed in Section 6. The bivariate proportional reversed hazard (BPRP) family of distributions is discussed in Section 7. A numerical study is discussed in Section 8. Finally, conclude the paper in Section 9.
2 Baseline Distributions
Some baseline distributions with the interesting properties will presented below.
-
i)
Exponential Distribution
If a continuous random variable X follows the exponential distribution then the pdf, survival function, hazard function, and cumulative hazard function are respectively:
-
ii)
Inverse Exponential Distribution
If a continuous random variable X follows the inverse exponential distribution then the cdf, pdf, reversed hazard function, and cumulative reversed hazard function are respectively:
-
iii)
Inverse Rayleigh Distribution
If a continuous random variable X follows the inverse Rayleigh distribution then the cdf, pdf, reversed hazard function, and cumulative reversed hazard function are respectively:
-
iv)
Uniform Distribution
If a continuous random variable X follows the uniform distribution then the cdf, pdf, hazard function,and cumulative hazard function are respectively:
-
v)
Inverse Uniform Distribution
The inverse uniform (IU) distribution is defined by using the transformation \( X=\frac{1}{T}-1 \) where T~U(0, 1). Then the cdf, pdf, reversed hazard function, and cumulative reversed hazard function for invers uniform distribution are respectively:
-
i)
Gompertz Distribution:
The pdf, survival function, hazard function, and cumulative hazard function of a continuous random variable X follows a Gompertz distribution are given respectively as:
-
ii)
Pareto Type I Distribution
The pdf, survival function, hazard function, and cumulative hazard function of a continuous random variable X follows a Pareto distribution are given respectively as:
3 Bivariate Reversed Hazard Power Parameter (BRPP) Family of Distributions
A reversed hazard power parameter model can be defined by adding a power parameter (α) to the formula F(x) = e−R(x), as following
where α > 0 and R(x) = − log F(x) is a cumulative reversed hazard function.
Where rB(.) and RB(.) are the baseline reversed hazard and cumulative reversed hazard functions respectively. Accordingly, the reversed hazard function for RPP family is given as
For more details in this manner [see Marshall and Olkin (2007), p257].
Now, the bivariate extension for this family is given as: Assume U1~RPP(α1), U2~RPP(α2) and U3~RPP(α3) and U′s are independent random variables. Let X1 = max (U1, U3) and X2 = max (U2, U3).
Then, (X1, X2) constitute a BRPP class of distributions denoted by BRPP(α1, α2, α3) with the following cdf and pdf
where x3 = min(x1, x2).
The joint cdf of BRPP models can be stretching in the following form
Where
Accordingly, the joint pdf of BRPP model can be obtained by the following proposition.
Proposition 1
Assume (X1, X2)~BRPP(α1, α2, α3) with the cdf FBRPP(x1, x2) defined in (3.4). Then the joint pdf for this class denoted by fBRPP(x1, x2) is given as
Where
and \( {f}_3(x)={\alpha}_3{r}_B(x){\left[{R}_B(x)\right]}^{\alpha_3-1}\exp \left\{-{\left[{R}_B(x)\right]}^{\alpha_1}-{\left[{R}_B(x)\right]}^{\alpha_2}-{\left[{R}_B(x)\right]}^{\alpha_3}\right\} \).
Proof
Let \( {x}_1<{x}_2 \). In this case FBRPP(x1, x2) in (3.4) becomes
Hence, by differentiation we get \( {f}_1\left({x}_1,{x}_2\right)=\frac{\partial^2{F}_1\left({x}_1,{x}_2\right)}{\partial {x}_1\partial {x}_2} \).
Similarly for x1 > x2 we can get the expression of f2(x1, x2) by the mixed derivatives \( \frac{\partial^2{F}_2\left({x}_1,{x}_2\right)}{\partial {x}_1\partial {x}_2} \) and hence \( {f}_2\left({x}_1,{x}_2\right)=\frac{\partial^2{F}_2\left({x}_1,{x}_2\right)}{\partial {x}_1\partial {x}_2} \).
But the expression of f3(x) can not be obtained by the similar manner. For this reason the following identity will be used
One can verify that
And
Since
Then, from (3.6) and (3.7) we can readily obtain
Which completes the proof.
The joint reversed hazard function of (X1, X2)~BRPP(α1, α2, α3) is obtained as follows
Where
and \( {r}_3(x)={r}_{RPP}\left(x;{\alpha}_3\right)={\alpha}_3{r}_B(x){\left[{R}_B(x)\right]}^{\alpha_3-1} \).
3.1 Marginal and Conditional Densities
Assume (X1, X2)~BRPP(α1, α2, α3), then the marginal cdf and pdf of X1 and X2 are given respectively, as follows
Further, for (X1, X2)~BRPP(α1, α2, α3), the conditional density of X1i given X2j = x2j is given by
where
3.2 Absolutely Continuous BRPP Family of Distributions
An absolutely continuous BRPP (BRPPac) family of distributions will be introduced by removing the singular part and remaining only the absolutely continuous part.
A random vector (Y1, Y2) follows a BRPPac family if its pdf is given by
Where
and C is a normalizing constant. It will be denoted as (Y1, Y2)~BRPPac(α1, α2, α3).
It is easy to check that the marginal distributions in this case are not univariate RPP models.
3.3 Parameters Estimation for BRPP Family of Distributions
In this section the shape parameters of BRPP models are estimated based on MLE method. Assume that {(x11, x21), (x12, x22), …, (x1n, x2n) } be a complete random sample from BRPP(α1, α2, α3) family of distributions whose pdf and cdf are given in (3.5) and (3.4). Consider the following notation
The log-likelihood function of the sample of size n from BRPP(α1, α2, α3) is given by
Where \( \underset{\_}{\alpha }=\left({\alpha}_1,{\alpha}_2,{\alpha}_3\right) \),
Accordingly, the likelihood equations can be written as
Where
and \( \Psi \left({x}_{ki};{\alpha}_k,{\alpha}_3\right)=\frac{{\left[{R}_B\left({x}_{ki}\right)\right]}^{\alpha_j-1}\left[1+{\alpha}_j\log \left[{R}_B\left({x}_{ki}\right)\right]\right]}{\alpha_j{\left[{R}_B\left({x}_{ki}\right)\right]}^{\alpha_j-1}+{\alpha}_3{\left[{R}_B\left({x}_{ki}\right)\right]}^{\alpha_3-1}},k=1,2,j=1,2;i\ne j. \)
Consequently, the second derivatives are given as follows
Where
3.3.1 Asymptotic Confidence Intervals
The asymptotic variance-covariance matrix of \( {\hat{\alpha}}_1,{\hat{\alpha}}_2 \)and\( {\hat{\alpha}}_3 \) is obtained by inverting the Fisher information matrix with elements that are negatives of expected values of the second order derivatives of logarithms of the likelihood function. In the present situation, it seems appropriate to approximate the expected values by their maximum likelihood estimates. Accordingly; the asymptotic variance –covariance matrix can be written as follows
Where \( {I}_{i\mathrm{j}}=-{\left.\frac{\partial^2l\left(\underset{\_}{\alpha}\right)}{\partial {\alpha}_i\partial {\alpha}_j}\right|}_{\alpha =\hat{\alpha}} \)
Now, the asymptotic normality results will be stated to obtain the asymptotic confidence intervals of α1, α2 and α3. Under particular regularity conditions it can be stated as follows.
Where F−1 is the variance-covariance matrix, \( \hat{\alpha}=\left({\hat{\alpha}}_1,{\hat{\alpha}}_2,{\hat{\alpha}}_3\right) \) and α = (α1, α2, α3).
3.4 A New Bivariate Distributions Belongs to BRPP Class
-
i)
Bivariate Inverse Weibull Distribution
A new bivariate inverse Weibull distribution is obtained by substituting Eqs. (2.7)–(2.8) in Eqs. (3.4)–(3.5) as follows.
The joint cdf for the bivariate inverse Weibull (BIW) distribution is given as
It can be rewritten in the following form
where x3 = min(x1, x2)
The corresponding joint pdf is given as
where
-
ii)
Bivariate Generalized Invers Rayleigh Distribution
A new bivariate Generalized inverse Rayleigh distribution is defined by using Eqs. (2.11)–(2.12) in Eqs. (3.4)–(3.5) with the following joint cdf and pdf respectively
where x3 = min(x1, x2). and denoted by BGIR(α1, α2, α3, σ).
The joint cdf of BGIR model can be stretching in the following form
Where
Accordingly, the joint pdf of BGIR model can be obtained as
Where
-
iii)
Bivariate Generalized Inverse Uniform Distribution
A bivariate generalized inverse uniform distribution denoted by BGIU(α1, α2, α3) can be introduced by substituting Eqs. (2.15)–(2.16) in Eqs. (3.4)–(3.5) to get the joint cdf and pdf as follows
where x3 = min(x1, x2).
The joint cdf of BGIU model can be stretching in the following form
Where
Accordingly, the joint pdf of BGIU model can be obtained as
where
4 Bivariate Hazard Power Parameter (BHPP)Family of Distributions
The survival function S(.) and its corresponding cumulative hazard function H(.) can related by the following formula S(x) = e−H(x), ∀ x.
So, The hazard power parameter model can be defined as follows
The corresponding pdf is given by differentiating (4.1) as
Where hB(.) and HB(.) are the baseline hazard and cumulative hazard functions respectively. Accordingly, the hazard function for HPP family is given as
It is follows if hB increasing and α ≥ 1, then hHPP is increasing; if hB decreasing and \( 0<\alpha <1 \), then hHPP is decreasing
Now, to get the bivariate HPP class of distributions. Assume the univariate hazard power parameter model is denoted by HPP(α, Θ) where α is the hazard power parameter and Θ may be a vector of parameters for an underlying distribution. Now suppose that Ui~HPP(αi, Θ), i = 1, 2, 3 such that Ui′s are mutually independent random variables and define Xj = min (Uj, U3 ), j = 1, 2. Such that; Xj′s are dependent random variables. Hence BHPP model denoted by BHPP(α1, α2, α3) is defined with the following joint survival function.
where x3 = max(x1, x2).
The joint survival function of (X1, X2)~BHPP(α1, α2, α3) can be stretching in the following form
Where
Accordingly, the joint pdf of BHPP model can be obtained by the following proposition.
Proposition 2
Assume (X1, X2)~BHPP(α1, α2, α3) with the survival function SBHPP(x1, x2) defined in (4.4). Then the joint pdf for this family denoted by fBHPP(x1, x2) is given as
where
Proof
By Following the same idea as in Proposition 1 the pdf is derived
The joint hazard function of the dependent variables (X1, X2)~BHPP(α1, α2, α3) is obtained as follows
where
4.1 Marginal and Conditional Densities
Assume (X1, X2)~BHPP(α1, α2, α3).Then, the marginal survival functions and densities of X1 and X2 are given respectively, as follows
Further, for (X1, X2)~BHPP(α1, α2, α3), the conditional density of X1i given X2j = x2j is given by
where
4.2 Absolutely Continuous BHPP Family of Distributions
An absolutely continuous BHPP (BHPPac) family of distributions will be introduced by removing the singular part and remaining only the absolutely continuous part.
A random vector (Y1, Y2) follows a BHPPac family if its pdf is given by
where
and C is a normalizing constant. It will be denoted as (Y1, Y2)~BHPPac(α1, α2, α3).
Again the marginal distributions in this case are not univariate HPP models.
4.3 Parameter Estimation for BHPP Family of Distributions
Assume that {(x11, x21), (x12, x22), …, (x1n, x2n) } be a complete random sample from BHPP(α1, α2, α3) family of distributions whose pdf and survival function are given in Eqs. (4.5) and (4.4). Again, consider the following notation
The log-likelihood function of the sample of size n from BHPP(α1, α2, α3) is given by
Where \( \underset{\_}{\alpha }=\left({\alpha}_1,{\alpha}_2,{\alpha}_3\right) \),
for k = 1, 2.
Accordingly, the likelihood equations can be written as
Where
and \( \Psi \left({x}_{ki};{\alpha}_k,{\alpha}_3\right)=\frac{{\left[{H}_B\left({x}_{ki}\right)\right]}^{\alpha_k-1}\left[1+{\alpha}_k\log \left[{H}_B\left({x}_{ki}\right)\right]\right]}{\alpha_k{\left[{H}_B\left({x}_{ki}\right)\right]}^{\alpha_k-1}+{\alpha}_3{\left[{H}_B\left({x}_{ki}\right)\right]}^{\alpha_3-1}},k=1,2. \)
The second derivatives are given as follows
Where
The asymptotic variance-covariance matrix for the parameters of BHPP family of distributions can be obtained by using the above second derivatives and doing the same steps explained in Section (3.3).
4.4 A New Bivariate Distributions Belongs to BHPP Class
-
i)
Bivariate Weibull Distribution
Using Eqs. (2.3)-(2–4) in Eqs. (4.4)–(4.5). A new bivariate Weibull distribution denoted by BW(α1, α2, α3, λ) can be defined by the joint survival function
where x3 = max(x1, x2).
The joint survival function of BW model can be stretching in the following form
Where
Accordingly, the joint pdf of BW model can be obtained as
where
Surface plots of the joint pdf of the BW model are given in Fig. 1. Where the values of (α1, α2, α3, λ) are taken to be as follows a = (5,5,5,0.5), b = (2,3,10,0.6), c = (8,4,3,0.5) d = (2,2,2,1), e = (2,2.5,2,2) and f = (1,1,1,0.05)
-
ii)
Bivariate Generalized Gompertz (BGG) Distribution
3D plots for the pdf of the absolutely continuous part of the BW model
Using Eqs. (2.23)-(2–24) in Eqs. (4.4)–(4.5). A new bivariate generalized Gompertz distribution denoted by BGG(α1, α2, α3, λ, ξ) can be defined by the following joint survival function
where x3 = max(x1, x2).
Or, the joint survival function of BGG model can be written as
Where
The joint pdf of BGG model can be written as
where
Surface plots of the joint pdf of the BGG model are given in Fig. 2. Where the values of (α1, α2, α3, λ, ξ) are taken to be as follows a = (2, 2.5, 2, 1, 1), b = (2, 2, 2, 1, 0.1), c = (1.5, 1.5,1,1,0.2), d = (1.2,1.5,1.1, 1, 0.2), e = (1,1,1,1,0.2) and f = (1,2.4,1,1,1).
-
iii)
Bivariate Generalized Pareto (BGP) Distribution
3D plots for the pdf of the absolutely continuous part of the BGG model
Using Eqs. (2.27)-(2-28) in Eqs. (4.4)–(4.5). A BGP distribution denoted by BGP(α1, α2, α3, λ) can be introduced by the joint survival function
where x3 = max(x1, x2).
That is,
Where
The joint pdf of BGP model can be written as
where
Surface plots of the joint pdf of the BGP model are given in Fig. 3. Where the values of (α1, α2, α3, λ) are taken to be as follows a = (3,2,1, 0.5), b = (3,2,2,5), c = (3,2,1,0,05), d = (2,3,2,0.5) e = (3,2,1,5) and f = (1,1,1,0.5)
3D plots for the pdf of the absolutely continuous part of the BGP model
-
iv)
Bivariate Generalized Uniform (BGU) Distribution
Using Eqs. (2.15)-(2-16) in Eqs. (4.4)–(4.5). A new BGU distribution denoted by BGU(α1, α2, α3) can be introduced by the joint survival function
where x3 = max(x1, x2).
where
The joint pdf of BGU model can be written as
where
5 Bivariate Power Parameter Family of Distributions (BPP)
Let FB be a baseline cdf. Suppose that that FPPF(.; α) is defined in terms of FB by the formula
Then α is called a power parameter and {FPPF(.; α), \( \alpha >0 \)} is a power parameter family with underling distribution FB.
The corresponding pdf and hazard function is given respectively, as
Where fBand hB are a baseline pdf and hazard functions respectively.
The bivariate version correspondence to this family can introduced as follows Assuming that U1, U2 and U3 are mutually independent random variables such that
U1~PP(α1), U2~PP(α2) and U3~PP(α3). Define X1 = Max (U1, U3) and X2 = Max (U2, U3) then by using Eqs. (5.1) and (5.2), the bivariate Power parameter family of distributions denoted by BPP(α1, α2, α3) is defined by the following joint cdf
Where
And \( {F}_3(x)={F}_B\left({x}^{\alpha_1}\right)\ {F}_B\left({x}^{\alpha_2}\right){F}_B\left({x}^{\alpha_3}\right). \)
6 Bivariate Proportional Hazard Family of Distributions (BPHP)
Let SB be a baseline survival function with cumulative hazard function HB(x) = − log SB(x) Suppose that SFPF(.; α) is defined in terms of SB by the formula
In this case α is called a frailty parameter and {SFP(.; α), \( \alpha >0 \)} is a frailty parameter family, or alternatively, a proportional hazard family with underlying survival functionSB.
The corresponding pdf and hazard function is given respectively, as
Where fBand hB are a baseline pdf and hazard functions respectively.
The bivariate version correspondence to this family has introduced by Shoaee (2020) as follows:
Assuming that U1, U2 and U3 are mutually independent random variables such that
U1~UFP(α1), U2~UFP(α2) and U3~UFP(α3). Define X1 = min (U1, U3) and X2 = min (U2, U3) then by using Eqs. (6.1) and (6.2), the bivariate frailty parameter family of distributions(or bivariate proportional hazard models) denoted by BFP(α1, α2, α3) is defined by the joint survival and density functions respectively, as follows
and
Where
and \( {f}_3(x)={\alpha}_3{f}_B(x)\ {\left[{S}_B\left({x}_1\right)\right]}^{\alpha_1+{\alpha}_2+{\alpha}_3-1} \).
7 Bivariate Proportional Reversed Hazard Family of Distributions (BPRP)
Suppose that FRP(.; α) is defined in terms of FB by the formula
In this case α is called a resilience parameter and {FRP(.; α), \( \alpha >0 \)} is a resilience parameter family, or alternatively, a proportional reversed hazard family with baseline cdf FB.
The corresponding pdf and reversed hazard function is given respectively, as
Where fBand rB are a baseline pdf and reversed hazard functions respectively.
Kundu and Gupta (2010) introduced a bivariate proportional reversed hazard family of distributions with the joint cdf and pdf respectively, as follows
\( {F}_{BRP}\left({x}_1,{x}_2\right)={\left[{F}_B\left({x}_1\right)\right]}^{\alpha_1}{\left[{F}_B\left({x}_2\right)\right]}^{\alpha_2}{\left[{F}_B\left({x}_3\right)\right]}^{\alpha_3} \), such that x3 = min(x1, x2)
where
and \( {f}_3(x)={\alpha}_3{f}_B(x)\ {\left[{F}_B(x)\right]}^{\alpha_1+{\alpha}_2+{\alpha}_3-1} \).
8 Numerical Study
8.1 Simulation Study
As be mentioned above the BRPP and BHPP contain different distributions. So it is better to use the different distributions in both BRPP and BHPP families. Such as bivariate Weibull (BW), bivariate generalized Gompertz (BGG), bivariate generalized inverse uniform (BGIU) distributions. Also, the following algorithm can be used to simulate these families in general.
Algorithm to generate from BRPP models
-
Step 1. Generate U1, U2 and U3 from U(0, 1).
-
Step 2. Compute \( {\mathrm{Z}}_1={R}_B^{-1}\left({\left[-\log {U}_1\right]}^{1/{\alpha}_1}\right) \), \( {\mathrm{Z}}_2={R}_B^{-1}\left({\left[-\log {U}_2\right]}^{1/{\alpha}_2}\right) \),
and \( {\mathrm{Z}}_3={R}_B^{-1}\left({\left[-\log {U}_3\right]}^{1/{\alpha}_3}\right) \).
-
Step3. Obtain X1 = min (Z1, Z3) and X2 = min (Z2, Z3).
-
Step4. Define the indicator functions as
\( {\delta}_{1i}=\left\{\begin{array}{c}1;\kern1.25em {x}_{1i}<{x}_{1i}\\ {}0;\kern1em otherwise\end{array}\right.,\kern0.5em {\delta}_{2i}=\left\{\begin{array}{c}1;\kern1.25em {x}_{1i}>{x}_{1i}\\ {}0;\kern1em otherwise\end{array}\right.\kern0.5em and\kern0.5em {\delta}_{3i}=\left\{\begin{array}{c}1;\kern1.5em {x}_{1i}={x}_{1i}\\ {}0;\kern1em otherwise\end{array}\right.. \)
-
Step5. The corresponding sample size n must satisfy n = n1 + n2 + n3
Such that \( {n}_1=\sum \limits_{i=1}^n{\delta}_{1i},\kern1.5em {n}_2=\sum \limits_{i=1}^n{\delta}_{2i}\kern0.50em and\kern1.25em {n}_3=\sum \limits_{i=1}^n{\delta}_{1i}. \)
To generate from BHPP model apply the same steps from 1 to 5 except in step 2 the quantile function is exchanged to be
A Monte Carlo simulation study testing the performance of MLE for the BRPP and BHPP models parameters will be introduced in general and especially for BGIU model which defined by Eq. (3.12) and denoted by BGIU(α1, α2, α3) and belongs to the BRPP family, BGG model which defined by Eq. (4.9) and denoted by BGG(α1, α2, α3, λ, ξ) and belongs to the BHPP family, and BW model which defined by Eq. (4.8) and denoted by BW(α1, α2, α3, λ) and belongs to the BHPP family.
The evaluation of the MLE was performed based on the following quantities for each sample size: the mean of the MLEs (MLE) and the corresponding Mean Squared Error, (MSE). For different choices for the sample sizes and different sets of parameters real values which are as follows
Group1:
For BGG model (α1, α2, α3, λ, ξ) = (0.8, 0.7, 0.7, 0.2,0.002)
For BW model (α1, α2, α3, λ) = (0.2, 0.3, 0.5, 0.2)
For BGIU model (α1, α2, α3) = (1.3, 1.5, 1.2)
Group 2:
For BGG model (α1, α2, α3, λ, ξ) = (0.7, 0.7, 0.6, 0.3,0.002)
For BW model (α1, α2, α3, λ) = (0.8, 0.7, 0.7, 0.2)
For BGIU model (α1, α2, α3) = (2, 2.5, 2)
The results of these simulations are presented in Tables 1 and 2. These results are useful, and it is observed that in most of the cases as the sample size increases, the MSEs decrease. This represents that the MLEs are consistent.
8.2 Application to Real Data Sets
In this section, three real data sets will be examined. Here, these data will be fitted to seven sub models. Four of them belong to BHPP family namely: (i) bivariate Weibull (BW) distribution, which defined by Eq. (4.8) (ii) bivariate generalized Gompertz (BGG) distribution which defined by Eq. (4.9), (iii) bivariate generalized Pareto (BGP) distribution which defined by Eq. (4.10) and (iv) bivariate generalized uniform distribution (BGU) which defined by Eq. (4.11). And three of them belong to BRPP family namely (i) bivariate inverse Weibull (BIW) distribution which defined by Eq. (3.10), (ii) bivariate generalized invers Rayleigh (BGIR) distribution which defined by Eq. (3.11), and (iii) bivariate generalized inverse uniform (BGIU) distribution which defined by Eq. (3.12).
8.2.1 Data Set 1: UEFA Champion’s League Data
The data set has been obtained from Meintanis (2007) and represented in Table 3. He explained that: the data represent the football (soccer) data where at least one goal scored by the home team and at least one goal scored directly from a penalty kick, foul kick or any other direct kick (all of them together will be called as kick goal) by any team have been considered. Here X1 represents the time in minutes of the first kick goal scored by any team and X2 represents the first goal of any type scored by the home team. In this case all possibilities are open, for example X1 < X2 or X1 > X2 or X1 = X2 = X.
8.2.2 Data Set 2: Cholesterol Levels
This data set contains cholesterol levels at 5 and 25 weeks after treatment in 30 patients and represented in Table 4. Before analyzing this data, the transformation (X − 150)/100 is applied to all data, this transformation will not effect on the analysis and are for computational reasons only. This data set was used by Shoaee (2020). Again, in this case all possibilities are exist, i.e., X1 < X2 or X1 > X2 or X1 = X2 = X.
8.2.3 Data Set 3: Burr Data
The data set has been obtained from Shoaee (2020) and represented in Table 5. This dataset contains 50 observations on the burr. In the first component, the hole diameter is 12 mm and the sheet thickness is 3.15 mm. In the second component, the hole diameter is 9 mm and the sheet thickness is 2 mm. These two datasets are derived from two different machines. Also, in this case all possibilities are exist, for example X1 < X2 or X1 > X2 or X1 = X2 = X.
The marginal distributions of both families are fitted to each data set separately which are: Weibull (W), generalized Gompertz (GE), generalized Pareto (GP), generalized uniform (GU), inverse Weibull (IW), generalized inverse Rayleigh (GIR) and generalized inverse uniform (GIU). The MLEs, The Kolmogorov-Smirnov (K-S) distances between the fitted distribution and the empirical distribution function for X1 and X2 and their maximum are shown in Tables 6 and 7 separately.
Now, the three data sets will fit to the seven bivariate sub-models (BW, BGG, BGP, BGU, BIW, BGIR, BGIU) defined above, the MLEs, the standard error (SE) and the confidence intervals(CI) with confidence interval lengths (CIL) will be calculated for each data set and listed in Tables 8, 9, and 10. To compare these models with each other or with any other bivariate models that represent this data the Akaike information criterion (AIC), Bayesian information criterion (BIC), the consistent Akaike information criterion (CAIC) and Hannan-Quinn information criterion (HQIC) are calculated for each model and each data set and listed in Table 11.
9 Conclusion
In this study new bivariate families of distributions are proposed by adding an extra shape parameter to the base distributions by different manners using the hazard and reversed hazard functions. In most of the cases the joint probability distribution, joint distribution and joint hazard and joint reversed hazard functions can be expressed in compact forms. The maximum likelihood estimation is considered for the vector of the unknown parameters. A simulation study is performed to see the performances of the estimators. For illustrative purposes three data sets has been re-analyzed and the performances are quite satisfactory.
References
Kundu, D. and Gupta, R.D. (2010). A class of Bivariate Models with Proportional Reversed Hazard Marginals. Sankhya, Ser B, 72, 236-253.
Marshall, A.W. and Olkin, I. (1967). A multivariate exponential distribution. Journal of the American Statistical Association, 62, 30- 44.
Marshall, A.W and Olkin, I. (2007). Life Distributions –Structure of Nonparametric, Semiparametri and Parametric Families, Springer, New York,
Meintanis, S. G. (2007). Test of fit for Marshall-Olkin distributions with applications. Journal of Statistical Planning and inference, 137, 3954–3963.
Shoaee S (2020) On a New Class of Bivariate Survival Distributions Based on the Model of Dependent Lives and its Generalization, Applications and Applied Mathematics An International Journal 15(2):801 – 829.
Acknowledgments
The author is profoundly grateful to the anonymous referees and editor for their valuable suggestions and comments which improved the earlier draft of the article.
Funding
Not applicable
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of Interest/Competing Interests
The author declare that she have no conflict of interest
Rights and permissions
About this article
Cite this article
Muhammed, H.Z. On some Bivariate Semi Parametric Families of Distributions with a Singular Component. Sankhya A 85, 1260–1306 (2023). https://doi.org/10.1007/s13171-022-00288-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-022-00288-1