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Estimation of P(X ≤ Y ) for a Bivariate Weibull

Distribution

Nasser Davarzania, Firoozeh Haghighiband Ahmad Parsianb

a Department of Statistics,Faculty of Science, Payammenoor University, Tehran, IRAN.

b School of Mathematics, Statistics and Computer Science, University of Tehran,

Tehran, IRAN.

R´ esum´ e

Nous ´ etudions ici l’estimation de R = P(X ≤ Y ), quand X et Y sont des variables

al´ eatoires qui suivent la loi bivariate Weibull et X est censur´ ee ` a Y. On obtient la loi

marginale pour les donn´ ees observ´ ees et on en tire MLE, UMVUE et MME de R. Ainsi

qu’on obtient les estimateurs de Bayse sur la fonction SEL. On a effectu´ e une simulation

de Monte-Carlo pour comparer ces estimateurs.

Abstract

In this paper we deal with estimation of R = P(X ≤ Y ), when X and Y are random

variables from a bivariate weibull distribution and X is censored at Y. we obtain the

marginal distribution for observed data and drive MLE, UMVUE and MME of R. Also

we obtain Bayes estimators of R under squared error loss (SEL) function. Monte Carlo

simulations carried out to compare these estimators.

1Introduction

In the context of reliability the stress-strength model describes the life of a component

which has a random strength Y and is subjected to random stress X. The component

fails at the instant that the stress applied to it exceeds the strength and the component

will function satisfactorily whenever X ≤ Y . Thus R = P(X ≤ Y ) is a measure of

component reliability. Estimation of R = P(X ≤ Y ), when X and Y are random vari-

ables following a specified distributions has been discussed extensively in the literature

in both distribution free and parametric frame work. The problem of estimating R when

the stress and strength are dependent and follow bivariate normal has been discussed

by Enis and Geisser (1971) and Mukherjee and Saran (1985). Jana (1994) and Hanagal

(1995) discussed the estimation of R when (X,Y) follow bivariate exponential of Marshall-

Olkin(1967). Hanagal (1997) and Jeevanand (1997) discussed the estimation of R when

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(X,Y) has a bivariate Pareto distribution. Hanagal (1997) obtained an estimator of R

when the stress is censored at strength and (X,Y) has Marshall-Olkin’s Bivariate expo-

nential (MOBE) distribution. In this paper we consider estimation of R when the stress

is censored at strength and (X,Y) follows Marshall-Olkin’s bivariate weibull (MOBW)

distribution. Hanagal (1996) proposed a family of k-variate weibull distributions, which

includes the multivariate exponential distribution of Marshall-Olkin(1967) as an especial

case. When k=2, the bivariate weibull distribution has the following probability density

function:

where θ1,θ2,θ3,σ > 0. Notice that, the marginal distributions of X and Y are Weibull

with scale parameters θ∗

shape parameter. The parameter θ3quantifies the dependence between the two variables

(X,Y) and θ3= 0 implies that X and Y are independent . Also, notice that the model

introduced in (1) is not absolutely continuous and contains a singular component in which

P(X = Y ) > 0. When (X,Y) follows MOBW distribution with pdf as given in (1), it is

easy to verify that R as a measure of component reliability is given by

f(x,y) =

f1(x,y) = θ2(θ1+ θ3)σ2(xy)(σ−1)exp(−xσ(θ1+ θ3) − θ2yσ),

f2(x,y) = θ1(θ2+ θ3)σ2(xy)(σ−1)exp(−yσ(θ2+ θ3) − θ1xσ),

f◦(x,x) = θ3exp(−θxσ),

x > y

x < y

x = yθ = θ1+ θ2+ θ3,

(1)

1= θ1+ θ3and θ∗

2= θ2+ θ3, respectively and σ is the common

R =

θ∗

1

θ∗

1+ θ2

(2)

In section 2, we consider the dependent right censoring and suppose that X is censored

at Y and (X,Y) follows the MOBWdistribution.

for observed data. In section 3, we obtain MLE, MME and UMVUE of R . In section

4, we mainly consider the Bayesian inference on R under SEL function when the prior

distributions of θ∗

estimators cannot be expressed in an explicit form in both cases. We use Monte Carlo

Markov Chain (MCMC) method and Lindley’s approximation to obtain the desired Bayes

estimates and their estimated risks under SEL function. In section 5 we consider the

numerical methods for a comparison purpose of different methods.

We drive the marginal distribution

1and θ2are independent and σ is known. It is observed that the Bayes

2Preliminaries

Let (X1, Y1) , (X2, Y2) , ...,(Xn, Yn) be a random sample (rs) of size n from MOBW

distribution. If Xiis right censored at Yi, the observed data are Zi= min(Xi,Yi) and

Vi= u(Yi− Xi), i = 1,2,...,n, where

?

0

u(t) =

1if t ≥ 0

if t < 0,

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i.e., the potential data are (X1, Y1) , ...,(Xn, Yn), but the actual observed data are W =

(W1,...,Wn) where Wi= (Zi,Vi),i = 1,2,...,n. Notice that:

• Z1,...,Znare independent

• V1,...,Vnare independent.

we obtain pdf of Wi,i = 1,...,n, as

h(wi) = [h1(wi)]1−vi[h2(wi)]vi

= σ zσ−1

i

e−θzσ

iθ∗vi

1θ1−vi

2

.

(3)

Notice that Vi,i = 1,...,n has Bernouli distribution with parameter R=θ∗

1,...,n has an exponential distribution with failure rate θ, and Zσ

n ?

3MLs, MMs and UMVUs Estimators of θ∗

1

θand Zσ

i,i =

iand Viare independent.

Let V =

i=1Viand Zσ=

n ?

i=1Zσ

i.

1and θ2

Let (X1, Y1), (X2, Y2), ...,(Xn, Yn) be a random sample of size n from MOBW distri-

bution with the pdf defined in (1). Then the likelihood function based on actual observed

data w is given by

?n−1

i=1

and the log-likelihood function is

L(σ,θ∗

1,θ2) = σn

?

zi

?σ

e−θzσθ∗v

1θn−v

2

,(4)

l(σ,θ∗

1,θ2) = nlnσ + (σ − 1)

n ?

i=1lnzi− θzσ+ v lnθ1∗+ (n − v)lnθ2.

(5)

The MLEs of σ,θ∗

1and θ2, say ˆ σ,ˆθ∗

1andˆθ2respectively, can be obtained as

ˆθ∗

1(σ) =V

Zσ

,

ˆθ2(σ) =n − V

Zσ

,

ˆθ(σ) =

n

Zσ. (6)

It is observed that the MLE of σ cannot be expressed in an explicit form and it can be

obtained by numerical methods. So the MMEs of θ∗

ˆθ∗

R can be obtained from (2)

1and θ2can be obtained from (6) as

1(ˆ σ) andˆθ2(ˆ σ), respectively. Hence, using the invariance property of MLE’s the MLE of

R =

ˆθ∗

1(ˆ σ)

ˆθ∗

1(ˆ σ) +ˆθ2(ˆ σ)

=V

n.

(7)

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When σ is known, the MMEs of θ∗

the following equations

1,θ2,θ and R, say˜θ∗

1,˜θ2,˜θ and˜R, can be obtained from

Γ(1+1

σ)

θ

1

σ

=1

n

n ?

n.

i=1Zi

θ∗

1

θ= R =V

Therefore,

˜θ∗

1=

Γ(1 +1

n ?

σ)

1

n

i=1Zi

σ

V

n

,

˜θ2=

Γ(1 +1

n ?

σ)

1

n

i=1Zi

σ

n − V

n

,

˜θ =

Γ(1 +1

n ?

σ)

1

n

i=1Zi

σ

,

˜R =V

n.

(8)

We know that V ∼ Bin(n,R) so it is interesting to observe that MLE, MME, and UMVUE

of R are the same in both parametric and non-parametric cases and is equal toV

n.

4 Bayesian Estimation of R

In this section, we deal with Bayes estimation of R based on actual observed data w,

under SEL function when σ is known or unknown. We see that in both cases the Bayes

estimators cannot be obtained in a closed form, so we apply an MCMC methods and

Lindley’s approximation, for a numerical comparison purpose.

4.1Bayesian Estimation of R when σ is known

Suppose θ∗

sities.

1and θ2have independent gamma prior distributions, with the following den-

π1(θ∗

1) =θ∗α1−1

1

e−θ∗

Γ(α1)

e−θ2η2ηα2

Γ(α2)

1η1ηα1

1

,(9)

π2(θ2) =θα2−1

22

, (10)

where αi,ηi> 0, i = 1,2, and σ are known. The joint posterior density function of θ∗

and θ2is

1

π (θ2,θ∗

1|w) ∝ σn

?n−1

i=1

?

zi

?σ

θ∗v+α1−1

1

e−θ∗

1(zσ+η1)θn−v+α2−1

2

e−θ2(zσ+η2).(11)

From (11), it is easy to verify that

θ∗

1|w∼ Γ(v + α1, η1+ zσ)andθ2|w∼ Γ(α2+ n − v,η2+ zσ) . (12)

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Notice that, R is a function of θ∗

can easily obtain the posterior density function of R based on the actual observed data

w in the form

1and θ2, hence, using the transformation methods one

f (r|w) = C

rv+α1−1(1 − r)n−v+α2−1

[r(zσ+ η1) + (1 − r)(zσ+ η2)]n+α1+α2,(13)

where

C =

Γ(n + α1+ α2)

Γ(n − v + α2)Γ(v + α1)(zσ+ η1)v+α1(zσ+ η2)n−v+α2.

Hence, the Bayes estimator of R under SEL function is given by

δ (w) =

?∞

◦

rv+α1(1 − r)n−v+α2−1

[r(zσ+ η1) + (1 − r)(zσ+ η2)]n+α1+α2dr. (14)

Notice that, when η1= η2= (η), (14) reduces to

δ (w) =

v + α1

n + α1+ α2

Since, the obtained estimator is unique Bayes estimator hence, it is admissible.

We will employ an MCMC Method to obtain estimated risk and bias of R. Also, using

the Lindley’s approximation (1980) with the approach of Ahmad and et al.(1997), it can

be easily seen that the approximate Bayes estimate of R under SEL function is

ˆRB=ˆˆR[1+

ˆˆθ1ˆˆR

ˆˆθ∗

1

2

(n − v + α2+ 1)(v + α1+ 1)

(ˆˆθ2(n−v +α2−1)−ˆˆθ∗(v +α1−1))], (15)

where

ˆˆθ∗

1=v + α1− 1

zσ+ η1

,

ˆˆθ2=n − v + α2− 1

zσ+ η2

,

ˆˆR =

ˆˆθ2

ˆˆθ∗

1+ˆˆθ2

5Numerical Results

As we recognized, in both cases of known and unknown σ, the obtained estimators lead

to some computational complexities. An analytic calculations of estimators and their

risks for comparison is not possible. Obviously, one sample dose not tell us too much.

However, it is not difficult to carry out an empirical comparison. For this purpose a

simulation study was conducted to generate a sequences of independent observations

using SAS9 package. The desired Bayes estimates when σ is known are calculated by

Metropolis-Hasting algorithm (MCMC method) and Lindley’s approximation method.

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When σ is unknown we obtain Bayes estimate using Gibbs sampling technique. We

obtained estimates N = 104times and calculate Estimated Risks (ER) given by

ER

?ˆR

?

=1

N

N

?

i=1

?ˆRi− R

?2,

WhereˆRiis an estimate of R.

We take θ1= 4,θ2= 6,θ3= 2, σ = 2, and follow the following steps for calculating ER.

The results are tabulated in Table 1

From Table 1 some of the points are quite clear. Even for small sample size, the perfor-

mance of the estimates are quite satisfactory in terms of biases and estimated MSEs. It

observed that when n increases, the estimated MSEs for MLEs decrease. This verifies the

consistency property of MLEs. The performance of the Bayes estimates in both meth-

ods are also quite satisfactory and as the sample size increases, their estimated MSEs

decrease.

Table 1. Biases and MSEs of MLEs and both Lindley’s approximation and MCMC method to Bayes

estimates of R, when σ is known and θ1= 4,θ2= 6,θ3= 2 and σ = 2 for different values of α1,α2,η1

and η2.∗

BAYES ESTIMATES

α1= 0.6,η1= 0.1

α2= 0.6

α2= 1

η2= 0.1

η2= 1/6

0.034900.03586

(-0.1293)(-0.1383)

0.02797 0.02986

(-0.1329)(-0.1406)

0.025970.02708

(-0.1371)(-0.1419)

0.024410.02564

(-0.1377)(-0.1421)

0.023990.02450

(-0.1401)(-0.14202)

LINDLEYS’s APPROXIMATION

α1= 0.6,η1= 0.1

α2= 0.6

α2= 1

η2= 0.1

η2= 1/6

0.031680.02408

(-0.1232)(-0.0874)

0.027370.02246

(-0.1314) (-0.1093)

0.025730.02141

(-0.1365)(-0.1190)

0.024280.02101

(-0.1374)(-0.1240)

0.023910.02062

(-0.1399)(-0.1270)

α1= 1,η1= 1/6

α2= 0.6

η2= 0.1

0.02761

(-0.1172)

0.02480

(–0.1229)

0.02392

(-0.1305)

0.02268

(-0.1317)

0.02261

(-0.1352)

α1= 1,η1= 1/6

α2= 0.6

η2= 0.1

0.03176

(-0.1578)

0.0319

(-0.1493)

0.02958

(-0.1519)

0.02728

(-0.1489)

0.02654

(-0.1496)

n

MLE

α2= 1

η2= 1/6

0.02967

(-0.1175)

0.02610

(-0.1284)

0.02484

(-0.1340)

0.02378

(-0.1361)

0.02295

(-0.1367)

α2= 1

η2= 1/6

0.02830

(-0.1139)

0.02559

(-0.1271)

0.02462

(-0.1334)

0.02367

(-0.1358)

0.02288

(-0.1364)

100.04378

200.03143

300.2809

40 0.02590

50 0.02516

∗In each box the first row presents estimated MSE of the estimates and the second row is its biases

reported within parentheses.

Bibliographie

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Amer. Statist. Assn., 66, 162-186.

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[5] Hanagal, D. D. (1997), Estimation of reliability when stress is censored at strength.

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