    Bivariate Discrete Inverse Weibull Distribution

In this paper, we propose a new class of bivariate distributions, called the bivariate discrete inverse Weibull (BDsIW) distribution, whose marginals are discrete inverse Weibull (DsIW) distributions. Some statistical and mathematical properties are presented. The maximum likelihood method is used for estimating the model parameters. Simulations are presented to verify the performance of the direct maximum likelihood estimation. Finally, two real data sets are analyzed for illustrative purposes.

Authors

07/25/2018

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1 Introduction

The Weibull (W) distribution is one of the most popular and widely used distributions for failure time in reliability theory (see, Weibull (1951)). The cumulative distribution function (CDF) of W distribution is given by

 Π(x;ν,ζ)=1−e−νxζ;  x>0, (1)

where is scale parameter and is shape parameter. Clearly, the exponential (E) distribution and the Rayleigh (R) distribution are special cases for and respectively. Unfortunately, the shape of the hazard rate function (HRF) of W distribution can be only increasing, decreasing or constant. So, more modifications and generalizations of W distribution are presented in the statistical literature to describe various phenomena in different fields, because in many applications, empirical hazard rate curves often exhibit non-monotonic shapes such as a bathtub, unimodal and others. For example:

1. Keller et al. (1985) proposed inverse Weibull (IW) distribution. The CDF of IW distribution is given by

 Π(x;ν,ζ)=e−νx−ζ;  x>0. (2)
2. Lai et al. (2003) introduced modified Weibull (MW) distribution. The CDF of MW distribution is given by

 Π(x;ν,ζ,λ)=1−e−νxζeλx;  x>0, (3)

where is an accelerating parameter. The exponentiated MW distribution was proposed by Jalmar et al. (2008).

3. Bebbington et al. (2007) proposed flexible Weibull (FxW) distribution. The CDF of FxW distribution is given by

 Π(x;ν,γ)=1−e−eνx−γx;  x>0, (4)

where is scale parameter. The exponentiated FxW distribution was presented by El-Gohary et al. (2015a), the inverse flexible Weibull (IFxW) distribution was proposed by El-Gohary et al. (2015b) and the exponentiated of it was studied by El-Morshedy et al. (2017).

4. Cordeiro et al. (2013) introduced exponential-Weibull (E-W) distribution. The CDF of E-W distribution is given by

 Π(x;γ,ν,β)=1−e−γx−νxβ;  x>0, (5)

where is shape parameter.

5. Nadarajah et al. (2013) proposed exponentiated Weibull (EW) distribution. The CDF of EW distribution is given by

 Π(x;ν,ζ,η)=(1−e−(νx)ζ)η;  x>0, (6)

where is shape parameter.

6. El-Bassiouny et al. (2017) introduced exponentiated generalized Weibull-Gompertz (EGW-Gz) distribution. The CDF of EGW-Gz distribution is given by

 Π(x;ν,ζ,λ,η,ρ)=(1−e−νxζ(eλxη−1))ρ;  x>0, (7)

where is shape parameter. The mixure of 2-EGW-Gz distribution was studied by El-Bassiouny et al. (2016).

Moreover, some discrete versions of E, R, W distributions and its generalizations have been presented in the literature because in several cases, lifetimes need to be recorded on a discrete scale rather than on a continuous analogue. So, discretizing continuous distributions has received much attention in the literature. For example:

1. Toshio and Shunji (1975) introduced discrete Weibull (DsW) distribution. The probability mass function (PMF) of DsW distribution is given by

 π(x;θ,ζ)=θxζ−θ(x+1)ζ;  x∈N0={0,1,2,3,...}, (8)

where . Clearly, the discrete Rayleigh (DsR) distribution is a special case for which was presented by Dilip (2004).

2. Gómez-Déniz (2010) proposed generalization of geometric (GGo) distribution. The PMF of GGo distribution is given by

 π(x;θ,γ)=γθx(1−θ)(1−[1−γ]θx+1)(1−[1−γ]θx);  x∈N0. (9)

The GGo distribution reduces to geometric or discrete exponential (DsE) distribution when

3. Jazi et al. (2010) introduced DsIW distribution. The PMF of DsIW distribution is given by

 π(x;θ,ζ)=θ(x+1)−ζ−θx−ζ;  x∈N0. (10)
4. Vahid et al. (2013) proposed discrete generalized exponential type II (DsGE-T2) distribution. The PMF of DsGE-T2 distribution is given by

 π(x;θ,ζ)=(1−θx+1)ζ−(1−θx)ζ;  x∈N0. (11)
5. Vahid and Hamid (2015a) introduced exponentiated discrete Weibull (EDsW) distribution. The PMF of EDsW distribution is given by

 π(x;θ,ρ,ζ)=(1−θ(x+1)ρ)ζ−(1−θxρ)ζ;  x∈N0. (12)
6. Vahid et al. (2015b) proposed discrete beta exponential (DsBE) distribution. The PMF of DsBE distribution is given by

 π(x;θ,γ,ζ)=cθγ(x−1)(1−θx)ζ−1;  x∈N0−{0}, (13)

where .

On the other hand, in many practical situations, it is important to consider different bivariate continuous and discrete distributions that could be used to model bivariate lifetime data in many fields. So, several bivariate continuous and discrete distributions are available in the statistical literature. For example, Lee (1997), Karlis and Ntzoufras (2000), Wu and Yuen (2003), Yuen et al. (2006), Sarhan and Balakrishnan (2007), Kundu and Dey (2009), Morata (2009), Kundu and Gupta (2009), Ong and Ng (2013), Balakrishnan and Shiji (2014), Lee and Cha (2015), Rasool and Akbar (2016), Hiba (2016), El-Bassiouny et al. (2016), El-Gohary et al. (2016), Vahid and Kundu (2017), Mohamed et al. (2017), Kundu and Vahid (2018), El-Morshedy and Khalil (2018) among others. An excellent encyclopedic survey of various continuous and discrete bivariate distributions can be found in Balakrishnan and Lai (2009) and Johnson et al. (1997) respectively.

In this regard, we focus the aim of this paper on presenting a flexible discrete bivariate distribution called BDsIW distribution, which can be usefully applied not only by statisticians, but also by data analysis in many different disciplines, such as sports, engineering, and medical applications. The proposed discrete model can be obtained from 3-independent DsIW distributions by using the maximization method as suggested by Lee and Cha (2015). The main reasons for introducing BDsIW distribution are:

1. The proposed model is a very flexible bivariate discrete distribution, and its joint PMF can take different shapes depending on the parameter values.

2. The generation from the proposed model is straight forward. So, the simulation experiments can be performed quite conveniently.

3. The marginals of the proposed model are DsIW distributions. Hence, the marginals are able to analyze the hazard rates in the discrete case.

4. The DsE and DsR distributions are special cases from the proposed model.

2 The BDsIW Distribution and Its Statistical Properties

2.1 Definition and interpretations

Suppose DsIW(), DsIW() and DsIW() and they are independently distributed. If

, then we can say that the bivariate vector

has a BDsIW distribution with the parameter vector = . We will denote this discrete bivariate distribution by BDsIW. If BDsIW, then the joint CDF of for and for is given by

 FX1,X2(x1,x2;Ψ) =θ(x1+1)−ζ1θ(x2+1)−ζ2θ(x3+1)−ζ3 =FDsIW(x1;θ1,ζ) FDsIW(x2;θ2,ζ) FDsIW(x3;θ3,ζ) (14)

The marginal CDFs for BDsIW distribution can be represented as follows

 FXd(xd;θd,θ3,ζ)=P[max(Wd,W3)≤xd]=FDsIW(xd;θdθ3,ζ). (15)

The corresponding joint PMF of for is given by

 fX1,X2(x1,x2;Ψ)=⎧⎪ ⎪⎨⎪ ⎪⎩f1(x1,x2;Ψ)              ; \ 0

where

 f1(x1,x2;Ψ) =fDsIW(x1;θ1θ3,ζ)fDsIW(x2;θ2,ζ), f2(x1,x2;Ψ) =fDsIW(x1;θ1,ζ) fDsIW(x2;θ2θ3,ζ), f3(x;Ψ) =FDsIW(x;θ2,ζ)fDsIW(x;θ1θ3,ζ)−FDsIW(x−1;θ2θ3,ζ)fDsIW(x;θ1,ζ).

The expressions , and for can be easily obtained by using the relation

 fX1,X2(x1,x2;Ψ)=F(x1,x2;Ψ)−F(x1−1,x2;Ψ)−F(x1,x2−1;Ψ)+F(x1−1,x2−1;Ψ). (17)

Figure 1 shows the plots of the joint PMF of BDsIW distribution for different parameter values.

Figure 1. The scatter plots of the joint PMF for different parameter values.

From Figure 1, it is clear that the joint PMF can take different shapes depending on the model parameter values. Assume BDsIW,then the joint reliability function of can be expressed as

 RX1,X2(x1,x2;Ψ) =1−FX1(x1;θ1θ3,ζ) −FX2(x2;θ2θ3,ζ)+FX1,X2(x1,x2;Ψ) =⎧⎪ ⎪⎨⎪ ⎪⎩R1(x1,x2;Ψ)              ; \ 0

where

 R1(x1,x2;Ψ) =1−FDsIW(x1;θ1θ3,ζ)−F% DsIW(x2;θ2θ3,ζ)+FDsIW(x1;θ1θ3,ζ) FDsIW(x2;θ2,ζ), R2(x1,x2;Ψ) =1−FDsIW(x1;θ1θ3,ζ)−F% DsIW(x2;θ2θ3,ζ)+FDsIW(x1;θ1,ζ) FDsIW(x2;θ2θ3,ζ),

and

 R3(x;Ψ)  =1−FDsIW(x;θ1θ3,ζ)−FDsIW(x;θ2θ3,ζ)+FDsIW(x;θ1θ2θ3,ζ).

Moreover, the bivariate hazard rate function (BHRF) of can be represented as

 rX1,X2(x1,x2;Ψ)=⎧⎪ ⎪⎨⎪ ⎪⎩r1(x1,x2;Ψ)              ; \ 0

where = Figure 2 shows the plots of the BHRF of BDsIW distribution for different parameter values.

Figure 2. The scatter plots of the BHRF for different parameter values.

From Figure 2, it is clear that the joint BHRF can take different shapes depending on the parameter values. Assume , then the HRF of the conditional distribution given is given by

 r∗(X1|X2>x2)=ζ(x1+1)−ζ−1R1(x1,x2;Ψ){FDsIW(x2;θ2,ζ)−1}FDsIW(x1;θ1θ3,ζ)ln(θ1θ3), (20)

and the HRF of the conditional distribution given is given by

 r∗(X2|X1>x1)=ζ(x2+1)−ζ−1F%DsIW(x2;θ2,ζ)R1(x1,x2;Ψ){FDsIW(x1;θ1θ3,ζ)ln(θ2)−FDsIW(x2;θ3,ζ)ln(θ2θ3)}. (21)

Similarly, when , then

 (22)

and

 (23)

On the other hand, assume a parallel system contains 2-component. Then, we can defined the BHRF as a vector which is useful to measure the total life span of a 2-component as follows

 r(x––)=(r(x),r12(x1|x2),r21(x2|x1)), (24)

where gives the HRF of the system using the information that the 2-component has survived beyond , gives the HRF span of the first component given that it has survived to an age and the other has failed at . Similar argument holds for , (see Cox (1972)). If BDsIW, then

 r(x)|X=min(X1,X2) =FDsIW(x−1;θ3,ζ)R3(x;Ψ)[−FDsIW(x−1;θ1,ζ)−FDsIW(x−1;θ2,ζ)+FDsIW(x−1;θ1θ2,ζ)] +FDsIW(x;θ3,ζ)R3(x;Ψ)[FDsIW(x;θ1,ζ)+FDsIW(x;θ2,ζ)−FDsIW(x;θ1θ2,ζ)],
 r12(x1|x2)|X1>X2=ζ(x1+1)−ζ−1[1−FDsIW(x1;θ1,ζ)]−1ln(θ1),

and

 r21(x2|x1)|X1

The following shock model and maintenance model interpretations can be provided for BDsIW distribution.

1. Shock model: Consider a system has 2-component, and it is assumed that the amount of shocks is measured in a discrete unit. Each component is subjected to individual shocks say and respectively. The system faces an overall shock , which is transmitted to both the component equally, independent of their individual shocks. So, the observed shocks at the 2- component are and respectively.

2. Maintenance model: Consider a system has 2-component, and it is assumed that each component has been maintained independently and also there is an overall maintenance. Due to component maintenance, assume the lifetime of the individual component is increased by amount and because of the overall maintenance, the lifetime of each component is increased by amount. Here, , and are all measured in a discrete unit. So, the increased lifetimes of the 2-component are and respectively.

2.2 Some statistical properties

Assume BDsIW, then and are positive quadrant dependent (PQD) where

 FX1,X2(x1,x2;Ψ)≥FX1(x1;θ1,θ3,ζ)FX2(x2;θ2,θ3,ζ). (25)

Furthermore, for every pair of increasing functions and , we get ; see for example Nelsen (2006). Let us recall that, the function , is said to have the total positivity of order two () property if satisfies

 k(u1,v1)k(u2,v2)≥k(u2,v1)k(u1,v2), (26)

for all . It is consider a very strong and an important property in lifetime testing, see for example Hu et al. (2003). Assume and from BDsIW, then the joint reliability function of satisfies the property where

 RX1,X2(x11,x21)RX1,X2(x12,x22)RX1,X2(x12,x21)RX1,X2(x11,x22)≥1. (27)

Similarly, when etc. Now, we present some statistical properties of the proposed model in a form of results.

Result 1. If the bivariate vector has the BDsIW, then

1. DsIW

2. The stress-strenght probability is given by

 P[X1
3. The median of and is given by

 MXd={logθdθ3U}1ζ−1; d=1,2, (29)

where  has a uniform distribution.

4. The coefficient of median correlation between and is given by

 MX1,X2=⎧⎨⎩4FDsIW(MX1;θ1θ3,ζ) FDsIW(MX2;θ2,ζ)−1        %; x1
5. The conditional PMF of given , is given by

 fX1∣X2=x2(x1∣x2)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩f(1)X1∣X2=x2(x1∣x2)    if  0

where

 f(1)X1∣X2=x2(x1 ∣x2)=fDsIW(x1;θ1θ3,ζ)fDsIW(x2;θ2,ζ)fDsIW% (x2;θ2θ3,ζ), f(2)X1∣X2=x2(x1 ∣x2)=fDsIW(x1;θ1,ζ),

and

6. The conditional CDF of given , is given by

 FX1∣X2=x2(x1∣x2)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩FDsIW(x1;θ1θ3,ζ) FDsIW(x2;θ3,ζ)      if  0

Proof. The proofs are quite standard and the details are avoided.

Result 2. Assume BDsIW for and they are independently distributed. If

 Zs=max(X1s,X2s,...,Xns);s=1,2⟹(Xi1,Xi2)∼BDsIW(n∏i=1θi1,n∏i=1θi2,n∏i=1θi3,ζ).

Proof. It is easy to proof that using the joint CDF.

Result 3. If the bivariate vector BDsIW, then the joint probability generating function (PGF) of and can be written as infinite mixtures,

 GX1,X2(y1,y2) =∞∑j=0j−1∑i=0[(θ1θ3)(i+1)−ζ−(θ1θ3)i−ζ][θ(j+1)−ζ2−θj−ζ2]yi1yj2 +∞∑i=0θ(i+1)−ζ2[(θ1θ3)(i+1)−ζ−(θ1θ3)i−ζ]yi1yi2 −∞∑i=0(θ2θ3)(i+1)−ζ[θ(i+1)−ζ1−θi−ζ1]yi1yi2;    |y1|,|y2|<1. (33)

Proof. The proof can be easily obtained by using the fact that

 GX1,X2(y1,y2)=E(yX11yX22)=∞∑j,i=0P[X1=i,X2=j]yi1yj2.

Hence, different moments and product moments of BDsIW distribution can be obtained, as infinite series, using the joint PGF.

3 Statistical Inference

3.1 Maximum likelihood estimation (MLE)

In this section, we use the maximum likelihood method to estimate the unknown parameters and of BDsIW distribution. Suppose that, we have a sample of size , of the form from BDsIW distribution. We use the following notations: and Based on the observations, the likelihood function is given by

 l(Ψ)=n1∏j=1f1(x1j,x2j)n2∏j=1f2(x1j,x2j)n3∏j=1f3(xj). (34)

The log-likelihood function becomes

 L(Ψ) =n1∑j=1ln(Φ1(x1j;θ1θ3,ζ))+n1∑j=1ln(Φ1(x2j;θ2,ζ))+n2∑j=1ln(Φ1(x1j;θ1,ζ))+n2∑j=1ln(Φ1(x2j;θ2θ3,ζ)) +n3∑j=1ln([θ2](xj+1)−ζΦ1(xj;θ1θ3,ζ)−[θ2θ3](xj)−ζΦ1(xj;θ1,ζ)), (35)

where The MLEs of the model parameters can be obtained by computing the first partial derivatives of Equation (35) with respect to and and then putting the results equal zeros. We get the likelihood equations as in the following form

 ∂L∂θ1 =n1∑j=1θ3Φ2(x1j+1;θ1θ3,ζ)−θ3Φ2(x1j;θ1θ3,ζ)Φ1(x1j;θ1θ3,ζ)+n2∑j=1Φ2(x1j+1;θ,ζ)−Φ2(x1j;θ,ζ)Φ1(x1j;θ1,ζ)+ n3∑j=1θ3θ(xj+1)−ζ2[Φ2(xj+1;θ1θ3,ζ)−Φ2(xj;θ1θ3,ζ)]−(θ2θ3)(xj)−ζ[Φ2(xj+1;θ1,ζ)−Φ2(xj;θ1,ζ)]θ(xj+1)−ζ2Φ1(xj;θ1θ3,ζ)−(θ2θ3)(xj)−ζΦ1(xj;θ1,ζ), (36)
 ∂L∂θ2 =n1∑j=1Φ2(x2j+1;θ2,ζ)−Φ2(x2j;θ2,ζ)Φ1(x2j;θ2,ζ)+n2∑