Estimation R=Pr(Y>X) for a Family of Lifetime Distributions by Transformation Method

Surinder Kumar and Prem Lata Gautam*

Department of Statistics, School for Physical and Decision Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow, India

E-mail: surinderntls@gmail.com; premgautm61@gmail.com

*Corresponding Author

Received 15 March 2021; Accepted 15 July 2021; Publication 23 August 2021

Abstract

For a Family of lifetime distributions proposed by Chaturvedi and Singh (2008) [6]. The problem of estimating R(t)=P(X>t), which is defined as the probability that a system survives until time t and R=P(Y>X), which represents the stress-strength model are revisited. In order to obtain the maximum likelihood estimators (MLE’S), uniformly minimum variance unbiased estimators (UMVUS’S), interval estimators and the Bayes estimators for the considered model. The technique of transformation method is used.

Keywords: Family of lifetime distributions, uniformly minimum variance unbiased estimator, maximum likelihood estimator, confidence interval, bayes estimator.

1 Introduction

The reliability of an item or system can be defined as a function of time ‘t’ i.e, R(t)=P(X>t), which defines the failure free operation of items/components until time ‘t’. One another important measure of reliability under the stress-strength model is R=Pr(Y>X), which represents the reliability of an item or system for the random strength Y and random stress X.

A lot of work has been done in the literature on the point estiamtion of R. For a brief review literature one may refer to Pugh (1963) [12], Basu (1964) [3], Church and Harris (1970) [8], Enis and Geisser (1971) [10], Downton (1973) [9], Tong (1974) [19], Kelly et al. (1976) [11], Sinha and Kale (1980) [15], Sathe and Shah (1981) [14], Chao (1982) [4], Awad and Gharraf (1986) [2], Chaturvedi and Surinder (1999) [7], Rezaei et al. (2010) [13], Chaturvedi and Pathak (2012) [5], Surinder and Mayank(2014) [18], Surinder and Mukesh (2015) [16] and Surinder and Mukesh (2016) [17].

2 The Family of Lifetime Distributions

Chaturvedi and Singh (2008) [6] derived a family of lifetime distributions with the help of Weibull distribution. Let the random variable X follows a family of lifetime distributions, then the pdf is presented as

f(x;a,λ,θ¯)=G(x;a,θ¯)λexp(-G(x;a,θ¯)λ);x>a0,λ>0 (1)

Here, G(x;a,θ¯) is a function of x and may also depend on the parameters a and θ¯. θ¯ may be vector valued. G(x;a,θ¯) represents the derivative of G(x;a,θ¯) with respect to x.

The presented model (1) covers the following lifetime distributions as specific cases:

1. For G(x;a,θ¯)=x and a=0, we get the one-parameter exponential distribution.

2. For G(x;a,θ¯)=xp,(p>0) and a=0, we get the Weibull distribution.

3. For G(x;a,θ¯)=x2 and a=0, we get the Rayleigh distribution.

4. For G(x;a,θ¯)=log(1+xb),b>0 and a=0, we get the Burr distribution.

5. For G(x;a,θ¯)=log(xa), we get the Pareto distribution.

6. For G(x;a,θ¯)=log(1+xν),ν>0 and a=0, we get the Lomax distribution.

7. For G(x;a,θ¯)=log(1+xbν),b>0,ν>0 and a=0, we get the Burr distribution with scale parameter ν(>0).

8. For G(x;a,θ¯)=xγexp(νx),γ>0,ν>0 and a=0, we get the modified Weibull distribution.

9. For G(x;a,θ¯)=(x-a)+νλlog(x+νa+λ),ν>0,λ>0, we get the generalised Pareto distribution.

10. For G(x;a,θ¯)=bx+θ2x2,θ>0,b>0 and a=0, we get the linear exponential distribution.

11. For G(x;a,θ¯)=(1+xb)θ-1,θ>0,b>0 and a=0, we get the generalised power Weibull distribution.

12. For G(x;a,θ¯)=βb(ebx-1),β>0,b>0 and a=0, we get the Gompertz distribution.

13. For G(x;a,θ¯)=(exb-1),b>0 and a=0, we get the Chen distribution.

14. For G(x;a,θ¯)=(x-a), we get the two-parameter exponential distribution.

3 MLE of R=Pr(Y>X)

In the following theorem, MLE of R is derived through the transformation method

Theorem 1: The MLE of R is

R¨=T¯(y)T¯(y)+T¯(x) (2)

where, T¯(y)=1n2j=1n2H(yj;a2,θ2) and T¯(x)=1n1i=1n1G(xi;a1,θ1)

Proof: Let the random variable X follows a Family of lifetime distribution with pdf

f(x;a1,λ1,θ1)=G(x;a1,θ1)λ1exp(-G(x;a1,θ1)λ1);
x>a10,λ1>0 (3)

For the given equation (3), let us consider the transformation G(x;a1,θ1)=t. Then the distribution become

f(t;α)=1αexp(-tα) (4)

where, α=λ1.

Now, let us consider Y be a random variable with pdf

f(y;a2,λ2,θ2)=H(y;a2,θ2)λ2exp(-H(y;a2,θ2)λ2);
y>a20,λ2>0 (5)

Similarly, let us take the transformation z=H(y;a2,θ2) and β=λ2, we get

f(z;β)=1βexp(-zβ) (6)

Let t and z be two independent random variable which follows exponential distribution (4) and (6) with parameters α and β, respectively, where t=G(x;a1,θ1) and z=H(y;a2,θ2). The relaibility model is

R=Pr(z>t)=z=0t=0f(t;α)f(z;β)dtdz=z=0[1-exp(-zα)]1βexp(-zβ)dz

After solving, we get

R=ββ+α (7)

On replacing the α and β by their MLE’S i.e, α¨=t¯ and β¨=z¯. The MLE of R=Pr(z>t) is

z¯z¯+t¯

where, t¯=1n1i=1n1ti and z¯=1n2j=1n2zj. Finally, MLE of R is

R¨=T¯(y)T¯(y)+T¯(x)

where, T¯(y)=1n2j=1n2H(yj;a2,θ2) and T¯(x)=1n1i=1n1G(xi;a1,θ1).

Hence, the theorem follows.

1. Implication
Here, we consider the different cases for the distributions to obtain the MLE of R=Pr(Y>X) given in (2)

Values of parameters for The MLE of R=Pr(Y>X)

Distributions Values of Parameter
The one-parameter exponential distribution T¯(y)=1n2j=1n2yj and T¯(x)=1n1i=1n1xi
Weibull distribution T¯(y)=1n2j=1n2yjp and
T¯(x)=1n1i=1n1xip for p>0
Rayleigh distribution T¯(y)=1n2j=1n2yj2 and T¯(x)=1n1i=1n1xi2
Burr distribution T¯(y)=1n2j=1n2log(1+yjb) and T¯(x)=1n1i=1n1log(1+xib)
for b>0
Pareto distribution T¯(y)=1n2j=1n2log(yja2) and T¯(x)=1n1i=1n1log(xia1)
Lomax distribution T¯(y)=1n2j=1n2log(1+yjν) and T¯(x)=1n1i=1n1log(1+xiν)
for ν>0
Burr distribution with scale parameter ν(>0) T¯(y)=1n2j=1n2log(1+yjbν) and T¯(x)=1n1i=1n1log(1+xibν)
for b>0,ν>0
The modified Weibull distribution T¯(y)=1n2j=1n2yjγexp(νyj) and T¯(x)=1n1i=1n1xiγexp(νxi)
for γ>0,ν>0
The generalised T¯(y)=1n2j=1n2 [(yj-a2)+νλ2log(yj+νa2+λ2)]
Pareto distribution T¯(x)=1n1i=1n1 [(xi-a1)+νλ1log(xi+νa1+λ1)]
for λ1,λ2>0,  ν>0
The linear exponential distribution T¯(y)=1n2j=1n2[byj+θ22yj2] T¯(x)=1n1i=1n1[bxi+θ12xi2]
for θ1,θ2>0 and b>0
The generalised power Weibull distribution T¯(y)=1n2j=1n2[(1+yjb)θ2]-1 and T¯(x)=1n1i=1n1[(1+xib)θ1]-1
θ1,θ2>0 and b>0
The Gompertz distribution T¯(y)=1n2βb(ebΠj=1n2yj-1) and T¯(y)=1n1βb(ebΠi=1n1xi-1)
β,b>0
Chen distribution T¯(y)=1n2j=1n2(eyjb-1) and T¯(x)=1n1i=1n1(exib-1)
b>0
The two-parameter exponential distribution T¯(y)=1n2j=1n2(yj-a2) and T¯(x)=1n1i=1n1(xi-a1)

4 UMVUE of R=Pr(Y>X)

In the following theorem, UMVUE of R is derived through the transformation method

Theorem 2: The UMVUE of R is

R´={i=0n2-1(-1)iΓ(n1)Γ(n2)Γ(n2-i)Γ(n1+i)(T(x)T(y))i;T(x)<T(y)i=0n1-2(-1)iΓ(n1)Γ(n2)Γ(n2+i+1)Γ(n1-i-1)(T(y)T(x))i+1;T(x)T(y) (8)

where, T(y)=i=1n2H(yj;a2,θ2) and T(x)=i=1n1G(xi;a1,θ1).

Proof: Considering the transfomation G(x;a1,θ1)=t and z=H(y;a2,θ2), we have the transform Equations (4) and (6). To obtain the measure of reliabilIty estimate Pr(z>t), we required to obtain the UMVUE of f(t;α) and f(z;β) i.e, f´(t;α) and f´(z;β) respectively, which is given by

f´(t;α)=(n1-1)G(t;a1,θ1)n1t¯[1-G(t;a1,θ1)n1t¯]n1-2;
G(t;a1,θ1)<n1t¯ (9)

and

f´(z;β)=(n2-1)H(z;a2,θ2)n2z¯[1-H(z;a2,θ1)n2z¯]n2-2;
H(z;a2,θ1)<n2z¯ (10)

Now to obtain UMVUE of R we have,

R´=Pr(z>t)=t=0z=tf´(t;α)f´(z;β)dzdt

using (9) and (10)

R´=t=0n1t¯z=tn2z¯(n1-1)(n2-1)H(z;a2,θ2)G(t;a1,θ1)n1n2t¯z¯[1-G(t;a1,θ1)n1t¯]n1-2[1-H(z;a2,θ1)n2z¯]n2-2dzdt
let [1-H(z;a2,θ1)n2z¯]=w
=t=0min(n1t¯,n2z¯)(n1-1)(n2-1)G(t;a1,θ1)n1t¯[1-G(t;a1,θ1)n1t¯]n1-2
[wn2-1n2-1]01-H(t;a2,θ1)n2z¯dt
=t=0min(n1t¯,n2z¯)(n1-1)G(t;a1,θ1)n1t¯[1-G(t;a1,θ1)n1t¯]n1-2
[1-H(t;a2,θ1)n2z¯]n2-1dt
=t=0min(n1t¯,n2z¯)(n1-1)G(t;a1,θ1)n1t¯[1-G(t;a1,θ1)n1t¯]n1-2
i=0n2-1(-1)i(n2-1i)[H(t;a2,θ1)n2z¯]idt

Now consider the case n1t¯<n2z¯. Let 1-G(t;a1,θ1)n1t¯=u, for solving the integral assuming G(t;a1,θ1)=H(t;a2,θ2) i.e., a1=a2 and θ1=θ2.

R´=01(n1-1)i=0n2-1(-1)i(n2-1i)[n1t¯(1-u)n2z¯]iun1-1du=i=0n2-1(-1)iΓ(n1)Γ(n2)Γ(n2-i)Γ(n1+i)(n1t¯n2z¯)i

In a same manner, we tackle the case when n1t¯>n2z¯:

R´=i=0n1-2(-1)iΓ(n1)Γ(n2)Γ(n2+i+1)Γ(n1-i-1)(n2z¯n1t¯)i+1

The UMVUE of R=Pr(Y>X) is obtained by substituting n2z¯=T(y)=j=1n2H(yj;a2,θ2) and n1t¯=T(x)=i=1n1G(xi;a1,θ1).

Hence, the theorem follows.

2. Implication
Here, we consider the different cases for the distributions to obtain the UMVUE of R=Pr(Y>X) given in (4)

Values of parameters for The UMVUE of R=Pr(Y>X)

Distributions Values of Parameter
The one-parameter exponential distribution T(y)=j=1n2yj and T(x)=i=1n1xi
Weibull distribution T(y)=j=1n2yjp and T(x)=i=1n1xip for p>0
Rayleigh distribution T(y)=j=1n2yj2 and T(x)=i=1n1xi2
Burr distribution T(y)=j=1n2log(1+yjb) and T(x)=i=1n1log(1+xib)
for b>0
Pareto distribution T(y)=j=1n2log(yja2) and T(x)=i=1n1log(xia1)
Lomax distribution T(y)=j=1n2log(1+yjν) and T(x)=i=1n1log(1+xiν)
for ν>0
Burr distribution with scale parameter ν(>0) T(y)=j=1n2log(1+yjbν) and T(x)=i=1n1log(1+xibν)
for b>0,ν>0
The modified Weibull distribution T(y)=j=1n2yjγexp(νyj) and T(x)=i=1n1xiγexp(νxi)
for γ>0,ν>0
The generalised Pareto distribution T(y)=j=1n2[(yj-a2)+νλ2log(yj+νa2+λ2)]
T(x)=i=1n1[(xi-a1)+νλ1log(xi+νa1+λ1)]
for λ1,λ2>0,  ν>0
The linear exponential distribution T(y)=j=1n2[byj+θ22yj2] T(x)=i=1n1[bxi+θ12xi2]
for θ1,θ2>0 and b>0
The generalised power T(y)=j=1n2[(1+yjb)θ2]-1 and T(x)=i=1n1[(1+xib)θ1]-1
Weibull distribution θ1,θ2>0 and b>0
The Gompertz distribution T(y)=βb(ebΠj=1n2yj-1) and T(x)=βb(ebΠi=1n1xi-1)
β,b>0
Chen distribution T(y)=j=1n2(eyjb-1) and T(x)=i=1n1(exib-1)
b>0
The two-parameter exponential distribution T(y)=j=1n2(yj-a2) and T(x)=i=1n1(xi-a1)

5 Confidence Interval of R=Pr(Y>X)

In the following theorem, confidence interval of R is derived through the transformation method

Theorem 3: The confidence interval of R=Pr(Y>X) is

P(n2R~cn1(1-R~)(1-c)+n2R~c<R<n2R~dn1(1-R~)(1-d)+n2R~d)=1-σ (11)

where, R¨=z¯z¯+t¯ and 0<c<d.

Proof: From the Theorem 1, the MLE of R is ββ+α or z¯z¯+t¯. As we know n1t¯ and n2z¯ follows Gamma distribution with parameters (α,n1) and (β,n2), respectively. For Confidence Interval of R, we must obtain the exact distribution of the variable

δ=αn1t¯αn1t¯+βn2z¯ (12)

Let ρ=αn1t¯ and ϱ=βn2z¯ and observe that ρ and ϱ have gamma distribution with the parameters (1,n1) and (1,n2) respectively. New set of varible is δ=ρρ+ϱ.

On taking ψ=ϱ and expressing the old variable in terms of new ones ρ=δψ(1-δ). The Jacobian of transformation is J=(1-δ)-2ψ. The joint pdf of δ and ψ

Pr(δ,ψ)=e-(ψ1-δ)ψn1+n2-1δn1-1Γ(n1)Γ(n2)(1-δ)n1+1 (13)

Intergrating out ψ, we have the maginal distribution of δ

Pr(δ)=[B(n1,n2)]-1δn1-1(1-δ)n2-1;0<δ<1

Here, δ has a beta distribution with the known parameters n1 and n2. So we have, for any 0<c<d

Pr(c<δ<d)=Id(n1,n2)-Ic(n1,n2) (14)

where, Ix(n1,n2)=[B(n1,n2)]-10xzn1-1(1-z)n2-1dz is the incomplete beta function. After calculation for the conection of δ and R¨, we have the pivotal quantity

δ=[1+n2R¨(1-R)n2R(1-R¨)]-1

where, R=ββ+α and R¨=z¯z¯+t¯.

If c and d in (14) are such that for a given σ

Id(n1,n2)-Ic(n1,n2)=1-σ

then,

P(c<[1+n2R¨(1-R)n2R(1-R¨)]-1<d)=1-σ (15)

After solving the equation (15) for R.

P(n2R~cn1(1-R~)(1-c)+n2R~c<R<n2R~dn1(1-R~)(1-d)+n2R~d)=1-σ

The above equation is valid for any values of n1 and n2, large or small.

Hence the theorem follows.

3. Implication
Here, we consider the different cases for the distributions to obtain the Confidence Interval of R=Pr(Y>X) given in (11)

Values of parameters for The Confidence Integral of R=Pr(Y>X)

Distributions Values of Parameter
The one-parameter exponential distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2yj and T¯(x)=1n1i=1n1xi
Weibull distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2yjp and T¯(x)=1n1i=1n1xip,
p>0
Rayleigh distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2yj2 and T¯(x)=1n1i=1n1xi2
Burr distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2log(1+yjb) and
T¯(x)=1n1i=1n1log(1+xib),b>0
Pareto distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2log(yja2) and
T¯(x)=1n1i=1n1log(xia1),b>0
Lomax distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2log(1+yjν) and
T¯(x)=1n1i=1n1log(1+xiν), for ν>0
Burr distribution with scale parameter ν(>0) R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2log(1+yjbν) and
T¯(x)=1n1i=1n1log(1+xibν),ν>0 and b>0
The modified Weibull distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2yjγexp(νyj) and
T¯(x)=1n1i=1n1xiγexp(νxi),ν>0 and γ>0
The generalised Pareto distribution R~=T¯(y)T¯(y)+T¯(y) T¯(y)=1n2j=1n2 [(yj-a2)+γλ2log(yj+νa2+λ2)] and
T¯(x)=1n1i=1n1 [(xi-a1)+γλ1log(xi+νa1+λ1)],ν>0 and γ>0
The linear exponential distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2[byj+θ22yj2] and
T¯(x)=1n1i=1n1[bxi+θ12xi2],θ1,θ2>0 and b>0
The generalised power R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2[(1+yjb)θ2-1] and
Weibull distribution T¯(x)=1n1i=1n1[(1+xib)θ1-1],θ1,θ2>0 and b>0
The Gompertz distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2[βb(ebyj-1)] and
T¯(x)=1n1i=1n1[βb(ebxi-1)],β>0 and b>0
Chen distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2(eyjb-1) and
T¯(x)=1n1i=1n1(exib-1),b>0
The two-parameter exponential distribution R~=T¯(y)T¯(y)+T¯(x) T¯(y)=1n2j=1n2(yj-a2) and
T¯(x)=1n1i=1n1(xi-a1),a1,a2>0

6 Bayes Estimator of R=Pr(Y>X)

In the following theorem, Bayes estimator of R is derived through the Transformation method

Theorem 4: The Bayes estimator of R is

Rˇ={μ*ξ*+μ*(η*ω*)-μ*F12(μ*+ξ*,μ*+1,μ*+ξ*+1;B),forB<1μ*ξ*+μ*(ω*η*)-ξ*F12(μ*+ξ*,ξ*,μ*+ξ*+1;B1-B),forB<-1 (16)

where F12(a,b,c;z) is the hypergeometric series and B=ω*-η*ω*<1.

Proof: Let us consider t¯ and z¯ be the independent samples from the pdfs (4) and (6). Here considering the conjugate prior, inverse gamma distributions for α and β with the parameters μ, η, and ξ, ω, respectively. Prior is

π(α,β)α-μ-1e(-ηα)β-ξ-1e(-ωβ);μ,η,ξ,β>0 (17)

The likelihood is

L(α,β|t¯,z¯)=α-n1β-n2exp[-(i=1n1tiα+j=1n2zjβ)] (18)

Applying Bayes formula and using (17) and (18). The posterior density of (α,β) is

π(α,β|t¯,z¯)α-μ-n1-1e-(η+n1t¯)αβ-ξ-n2-1e-(ω+n2z¯)β (19)

Evidently the posterior risk is also the product of gamma pdfs with the updated parameters

μ*=-(n1+μ),η*=η+n1t¯,ξ*=-(ξ+n2),ω*=ω+n2z¯

where, t¯ and z¯ are the sample means.

For posterior pdf of R, we consider a one-to-one transformation F:R=ββ+α,ϑR=α+β with the inverse Q:α=RϑR,β=R(1-ϑR). The Jacobian of transformation is ϑR. The joint posterior density of R and ϑR becomes

π*(R,ϑR|t¯,z¯)Rμ*-1(1-R)ξ*-1ϑRμ*+ξ*-1e-ϑRω*(1-BR);
0<R<1,ϑR>0 (20)

where B=ω*-η*ω*<1.

Intergrating the (20) for ϑR

πR(R|t¯,z¯)=CRRμ*-1(1-R)ξ*-1(1-BR)-(μ*+ξ*);0<R<1 (21)

where, CR is the normalizing coefficient. For the Baye estimator we have

Rˇ=RπR(R|t¯,z¯)dR (22)

Using the (21) and solving (22), we obtain the bayes estimator of R

Rˇ={μ*ξ*+μ*(η*ω*)-μ*F12(μ*+ξ*,μ*+1,μ*+ξ*+1;B),forB<1μ*ξ*+μ*(ω*η*)-ξ*F12(μ*+ξ*,ξ*,μ*+ξ*+1;B1-B),forB<-1

where, F12(a,b,c;z)=j=1a(a+1)(a+j-1)b(b+1)(b+j-1)c(c+1)(c+j-1)zjj! is the hypergeometric series.

For the Bayes estimator R¨, replacing the parameters as

μ*=-(n1+μ),η*=η+n1T¯(x),ξ*=-(ξ+n2),ω*=ω+n2T¯(y)

Hence, the theorem follows.

4. Implication
Here, we consider the different cases for the distributions to obtain the Bayes estimators of R=Pr(Y>X) given in (16)

Values of parameters for The Bayes estimators of R=Pr(Y>X)

Distributions Values of Parameter
The one-parameter exponential μ*=-(n1+μ),η*=η+n1x¯,
distribution ξ*=-(ξ+n2),ω*=ω+n2y¯
Weibull distribution μ*=-(n1+μ),η*=η+i=1n1xip,
ξ*=-(ξ+n2),ω*=ω+i=1n2yjp,p>0
Rayleigh distribution μ*=-(n1+μ),η*=η+i=1n1xi2,
ξ*=-(ξ+n2),ω*=ω+i=1n2yj2,p>0
Burr distribution μ*=-(n1+μ),η*=η+i=1n1log(1+xib),
ξ*=-(ξ+n2),ω*=ω+i=1n2log(1+yjb),b>0
Pareto distribution μ*=-(n1+μ),η*=η+i=1n1log(xia1),
ξ*=-(ξ+n2),ω*=ω+i=1n2log(yja2),a1,a2>0
Lomax distribution μ*=-(n1+μ),η*=η+i=1n1log(1+xiν) ,
ξ*=-(ξ+n2),ω*=ω+i=1n2log(1+yjν),ν,b>0
Burr distribution with scale parameter ν(>0) μ*=-(n1+μ),η*=η+i=1n1log(1+xibν) ,
ξ*=-(ξ+n2),ω*=ω+i=1n2log(1+yjbν),ν,b>0
The modified Weibull distribution μ*=-(n1+μ),η*=η+i=1n1xiγexp(νxi) ,
ξ*=-(ξ+n2),ω*=ω+i=1n2yjγexp(νyj),γ,ν>0
The generalised Pareto distribution μ*=-(n1+μ),η*=η+i=1n1[(xi-a1)+νλ1log(xi+νa1+λ1)],
ξ*=-(ξ+n2),ω*=ω+i=1n2[(yj-a2)+νλ2log(yj+νa2+λ2)],
γ,ν>0
The linear exponential distribution μ*=-(n1+μ),η*=η+i=1n1[bxi+θ12xi2],
ξ*=-(ξ+n2),ω*=ω+i=1n2[byj+θ22yj2],b>0
The generalised power μ*=-(n1+μ),η*=η+i=1n1[(1+xib)θ1-1],
Weibull distribution ξ*=-(ξ+n2),ω*=ω+i=1n2[(1+yjb)θ2-1],b>0
The Gompertz distribution μ*=-(n1+μ),η*=η+i=1n1[βb(ebxi-1)],
ξ*=-(ξ+n2),ω*=ω+i=1n2[βb(ebyj-1)],b>0
Chen distribution μ*=-(n1+μ),η*=η+i=1n1[exib-1],
ξ*=-(ξ+n2),ω*=ω+i=1n2[eyjb-1],b>0
The two-parameter exponential distribution μ*=-(n1+μ),η*=η+i=1n1(xi-a1),
ξ*=-(ξ+n2),ω*=ω+i=1n2(yj-a2),a1,a2>0

7 Discussion

The Family of lifetime distribution is used in order to obtained the MLES, UMVUES, Confidence intervals and Bayes estimators of R for the various distributions. Initially, the generalized expressions for obtaining the MLES, UMVUES, Confidence intervals and Bayes estimators of R are obtained, then the estimator of the corresponding distributions are simply obtained by just replacing their respective parameters. For example, consider the following examples:-

Example 1 – Consider the Weibull distribution
Let X1,X2,Xn be a random sample from WE(α,λ1) and Y1,Y2,Ym be a random sample from WE(α,λ2). Amiri et al. (2013) [1] obtained the MLE and UMVUE of R for Weibull distribution, which is given as

R¨=mj=1myjαni=1nxiα+mj=1myjα

and the UMVUE of R is

R´={1-i=0m-1(-1)iΓ(n)Γ(m)Γ(n+i)Γ(m-i)(t1t2)i;t1<t2j=0n-1(-1)jΓ(n)Γ(m)Γ(n-j)Γ(m+j)(t2t1)j;t1t2

where, t1=i=1nxiα and t2=j=1myjα are the sufficient statistics for the λ1 and λ2.

Example 2 – Consider the Burr distribution
Let X be a Burr random variable with parameters (p, b) and Y is another Burr random variable with parameters (a, b). Awad and Gharraf (1986) [2] obtained the MLE and UMVUE of R for Burr distribution, which is given as

R¨=11+nmj=1mlog(1+yjb)j=1nlog(1+xjb)

and the UMVUE of R is

R´={j=0m-1(-1)j(m-1)!(n-1)!(m-1+j)!(n-1-j)!i=1mvii=1nwi(i=1mvii=1nwi)j;1-j=0m-1(-1)j(m-1)!(n-1)!(m-1-j)!(n-1+j)!i=1mvi>i=1nwi(i=1nwii=1mvi)j;

where, i=1nwi=j=1nlog(1+xjb) and i=1mvi=j=1mlog(1+yjb)

Example 3 – Consider the generalized Pareto distribution
Suppose X1,X2,Xn be a random sample from GP(α,λ) and Y1,Y2,Yn be a random sample from GP(β,λ). Rezaei et al. (2010) [13] obtained the MLE and UMVUE of R for generalized Pareto distribution, which is given as

R¨=mj=1mln(1+λyj)ni=1nln(1+λxi)+mj=1m(1+λyj)

and the UMVUE of R is

R´={1-i=0m-1(-1)i(m-1)!(n-1)!(m-i-1)!(n+i-1)!(T1T2)i;T1T2i=0n-1(-1)i(m-1)!(n-1)!(m+i-1)!(n-i-1)!(T2T1)i;T2T1

where, T1=i=1nln(1+Xi) and T2=i=1mln(1+Yi)

Remarks: All the above Example 1–3 are the specific cases of our generalized expressions. Thus, in this study we have suggested a very simple and approved method i.e, transformation method for obtaining the MLES, UMVUES, Confidence intervals and Bayes estimators of R for the different distributions.

References

[1] Amiri, N., Azimi, R., Yaghmaei, F. and Babanezhad, M. 2013: Estimation of stress-strength parameter for two-parameter weibull distribution. Int. J. of Adanced Stat. and prob., 1(1):4–8.

[2] Awad, A. M. and Gharraf, M. K. 1986: Estimation of P(Y<X) in the Burr case, A Comparative Study. Commun. Statist. – Simul., 15(2):389–403.

[3] Basu, D. 1964: Estimates of reliability for some distributions useful in life testing. Technometrics, 6:215–219.

[4] Chao, A. 1982: On comparing estimators of P(X>Y) in the exponential case. IEEE transactions on reliability, 31:389–392.

[5] Chaturvedi, A. and Pathak, A. 2012: Estimation of the reliability functions for exponentiated Weibull distribution. J. Stat. Appl., 7:1–8.

[6] Chaturvedi, A. and Singh, K. G. 2008: A family of lifetime distributions and related estimation and testing procedures for the reliability function. J. Appl. Stat. Sci., 16(2):35–50.

[7] Chaturvedi, A. and Surinder, K. 1999: Further remarks on estimating the reliability function of exponential distribution under Type-I and Type-II censorings. Brazilian Journal of Probability and Statistics, 13:29–39.

[8] Church, J. D. and Harries, B. 1970: The estimation of reliability from stress-strength relationships. Technometrics, 12:49–54.

[9] Downton, F. 1973: The estimation of Pr(Y<X) in the normal case. Technometrics, 15:551–558.

[10] Enis, P. and Geisser, S. 1971: Estimation of the probability that (Y>X). J. Amer. Statist. Asso., 66:162–168.

[11] Kelly, G. D., Kelly., J. A. and Schucany, W. R. 1976: Efficient estimation of P(Y<X) in the exponential case. Technometrics, 18:359–360.

[12] Pugh, E. L. 1963: The best estimate of reliability in the exponential case. Operations Research, 11:57–61.

[13] Rezaei, S., Tahmasbi, R. and Mahmoodi, M. 2010: Estimation of P(Y<X) for generalized Pareto distribution. J. Stat. Plan Inference, 140:480–494.

[14] Sathe, Y. S. and Shah, S. P. 1981: On estimating P(X<Y) for the exponential distribution. Commun. Statist. Theor. Meth., A10:39–47.

[15] Sinha , S. K. and Kale, B. K. 1980: Life testing and Reliability Estimation. Wiley Eastern Ltd., New Delhi.

[16] Surinder, K. and Kumar, M. 2015: Study of the Stress-Strength Reliability among the Parameters of Generalized Inverse Weibull Distribution. Intern. Journal of Science, Technology and Management, 4:751–757.

[17] Surinder, K. and Kumar, M. 2016: Point and Interval Estimation of R=P(Y>X) for Generalized Inverse Weibull Distribution by Transformation Method. J. Stat. Appl. Pro. Lett., 3:1–6.

[18] Surinder, K. and Mayank, V. 2014: On the estimation of R=P(Y>X) for a class of Lifetime Distributions by Transformation Method. J. Stat. Appl. Pro., 3(3):369–378.

[19] Tong, H. 1974: A note on the estimation of P(Y<X) in the exponential case. Technometrics, 16:625.

Biographies

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Surinder Kumar, Head, Dept. of Statistics, BBAU (A central University), Lucknow – India. He is having 26 years research experience in various research fields of Statistics such as Sequential Analysis, Reliability Theory, Business Statistics and Bayesian Inference. Prof. Kumar has published more than 60 research publications in various journals of national and international repute.

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Prem Lata Gautam, Dept. of Statistics, BBAU (A Central University) Lucknow, India. She has research experiences of 6 years and has also published 6 research articles in various reputed journals in the field of Sequential analysis, Bayesian estimation and Reliability theory and wholesome knowledge of many softwares and language like R Software, Mathematica and Fortron.

Abstract

1 Introduction

2 The Family of Lifetime Distributions

3 MLE of R=Pr(Y>X)

4 UMVUE of R=Pr(Y>X)

5 Confidence Interval of R=Pr(Y>X)

6 Bayes Estimator of R=Pr(Y>X)

7 Discussion

References

Biographies