Adv Syst Sci Appl 2018; 4:39–51
http://ijassa.ipu.ru/index.php/ijassa/article/view/572

Estimation of Stress-Strength Reliability for Quasi Lindley
Distribution

M.M. Mohie El-Din1, A. Sadek 1∗, Shaimaa H. Elmeghawry2

1 Dep. of Mathematics, Faculty of Science, Al-Azhar University, Egypt
2 Faculty of Engineering, Benha University, Egypt

Received April 14, 2018; Revised October 17, 2018; Published December 31, 2018

Abstract: This paper discussed the problem of estimating of the stress-strength reliability R =
Pr(Y < X). It is assumed that the strength of a system X, and the environmental stress applied
on it Y, follow the Quasi Lindley Distribution(QLD). Stress-strength reliability is studied using
the maximum likelihood, and Bayes estimations. Asymptotic confidence interval for reliability is
obtained. Bayesian estimations were proposed using two different methods: Importance Sampling
technique, and MCMC technique via Metropolis-Hastings algorithm, under symmetric loss
function (squared error) and asymmetric loss functions (linex, general entropy). The behaviors
of the maximum likelihood and Bayes estimators of stress-strength reliability have been studied
through the Monte Carlo simulation study. Finally analysis of a real data set has also been
presented.

Keywords: Quasi Lindley distribution; Stress-strength reliability; Maximum likelihood
estimation; Asymptotic confidence interval; Bayesian estimation; Importance
sampling technique; MCMC technique via Metropolis-Hastings algorithm.

1. INTRODUCTION

The stress-strength models have been widely used for reliability design of systems. In these
models the reliability is defined as the probability that the strength is larger than the stress
R = Pr(Y < X). As the strength X is larger than the stress Y, the system work efficiently,
otherwise the the system fails. Estimation of stress-strength reliability was studied by several
authors, for example, stress-strength model and its generalizations has been discussed
in [10]. The estimation of R when X and Y are normally distributed was introduced by
Church and Harris [7]. Krishnamoorthy et al. [11] introduced an inference on reliability
in two-parameter exponential stress-strength model. Al-Mutairi et al. [1, 2] presented
the stress-strength reliability for Lindley and weighted Lindley distributions respectively.
Stress-strength reliability estimation for generalized Lindley distribution has been introduced
by Singh et al. [18]. Recentely Khan and Jan [9] studied the estimation of stress-strength
reliability model using finite mixture of two parameter Lindley distributions.

This paper is focused upon studying the problem of the estimation of the stress-strength
reliability for the Quasi Lindley Distribution (QLD) introduced by Shanker et al. [17] of
which the Lindley distribution is a particular case. We estimated the parameter of the stress-
strength reliability R using the maximum likelihood, and Bayesian estimation methods. We

∗Corresponding author: a sadek@azhar.edu.eg



40 M. MOHIE EL-DIN, A. SADEK, SHAIMAA ELMEGHAWRY

construct the asymptotic confidence interval of R based on the asymptotic distribution of
the MLE of R. In Bayesian estimation we introduced two sampling methods (importance
sampling and MCMC).
The QLD has the following probability density function (PDF):

f(x) =
θ

α + 1
(α + θx) e−θx , (1.1)

and the following cumulative distribution function (CDF):

F (x) = 1−
[(1 + α + θx)

α + 1
e−θx

]
, (1.2)

where;
x > 0, θ > 0, α > −1.

This paper is organized as follows. In Section 2, stress-strength reliability issue is studied
to obtain the reliability function of the parameters of QLD distribution. Maximum likelihood
estimation for stress-strength reliability is discussed in Section 3. Asympototic confidence
interval of reliability is proposed in section 4. In Section 5, a general procedure for deriving
the Bayesian estimator of reliability is introduced, wherein we applied the importance
sampling and MCMC techniques to compute the approximation of this estimator. Section
6 presented simulation study to investigate and compare the performance of each method of
estimation. Section 7 presented analysis of a real data set for illustrative purposes. Finally,
conclusions appear in Section 7.

2. STRESS-STRENGTH RELIABILITY

Assume X ∼ QLD(θ1, α1) and Y ∼ QLD(θ2, α2) are independent random variables with
PDF f(x) and g(y), respectively. Then the stress strength reliability can be obtained as:

R = Pr(Y < X).

=

∫ ∞
0

∫ x

0

f(x)g(y) dydx.

=

∫ ∞
0

f(x)G(x) dx.

=

∫ ∞
0

θ1(α1 + θ1x)

α1 + 1
e−θ1x

[
1−

[
(1 + α2 + θ2x)

α2 + 1
e−θ2x

]]
dx.

=

∫ ∞
0

θ1(α1 + θ1x)

α1 + 1
e−θ1x −

∫ ∞
0

θ1(α1 + θ1x)(1 + α2 + θ2x) e−(θ1+θ2)x

(α1 + 1)(α2 + 1)
dx.

= 1−
θ1
(
2θ1θ2 + (θ1 + θ2) (α2θ1 + α1θ2 + θ1) + α1 (α2 + 1) (θ1 + θ2)

2)
(α1 + 1) (α2 + 1) (θ1 + θ2)

3 . (2.3)

From Eq.(2.3), we noticed that R is a function of parameters Θ = (θ1, α1, θ2, α2). Therefore,
for maximum likelihood estimate (MLE) of R we need to obtain the MLEs of these
parameters.

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



ESTIMATION OF STRESS-STRENGTH RELIABILITY 41

3. MAXIMUM LIKELIHOOD ESTIMATION FOR RELIABILITY

Suppose that x1, x2, · · · , xn is random sample from QLD(θ1, α1), and y1, y2, · · · , ym is
random sample from QLD(θ2, α2), then the log likelihood function can be written as:

logL(x, y; Θ) = n log (θ1) +m log (θ2)− n log (α1 + 1)−m log (α2 + 1)

−θ1
n∑
i=1

xi − θ2
m∑
j=1

yj +
n∑
i=1

log (α1 + θ1xi) +
m∑
j=1

log (α2 + θ2yj) .

(3.4)

The MLE of Θ = (θ1, α1, θ2, α2) can be obtained as a solution of the following equations:

∂L

∂θ1
=
n

θ1
−

n∑
i=1

xi +
n∑
i=1

xi
α1 + θ1xi

= 0,

∂L

∂θ2
=
m

θ2
−

m∑
j=1

yj +
m∑
j=1

yj
α2 + θ2yj

= 0,

∂L

∂α1

= − n

α1 + 1
+

n∑
i=1

1

α1 + θ1xi
= 0,

and
∂L

∂α2

= − m

α2 + 1
+

m∑
j=1

1

α2 + θ2yj
= 0.

Solving these equations numerically using an iterative process as Newton Raphson to get
θ̂1, α̂1, θ̂2, α̂2, then the MLE of R can be obtained as following:

R̂ = 1−
θ̂1

(
2θ̂1θ̂2 +

(
θ̂1 + θ̂2

)(
α̂2θ̂1 + α̂1θ̂2 + θ̂1

)
+ α̂1 (α̂2 + 1)

(
θ̂1 + θ̂2

)
2
)

(α̂1 + 1) (α̂2 + 1)
(
θ̂1 + θ̂2

)3 . (3.5)

4. ASYMPTOTIC CONFIDENCE INTERVAL OF R

The asymptotic variance-covariance matrix of all parameters can be approximated by the
inverse of observed information matrix, and then derive the asymptotic distribution of R̂.
Based on the asymptotic distribution of R̂, we obtain the asymptotic confidence interval of R.
The Fisher information matrix of Θ = (θ1, α1, θ2, α2) is given as:

I(Θ) = −


E( ∂

2L
∂θ1

2 ) E( ∂2L
∂θ1∂α1

) E( ∂2L
∂θ1∂θ2

) E( ∂2L
∂θ1∂α2

)

E( ∂2L
∂α1∂θ1

) E( ∂
2L

∂α1
2 ) E( ∂2L

∂α1∂θ2
) E( ∂2L

∂α1∂α2
)

E( ∂2L
∂θ2∂θ1

) E( ∂2L
∂θ2∂α1

) E( ∂
2L

∂θ2
2 ) E( ∂2L

∂θ2∂α2
)

E( ∂2L
∂α2∂θ1

) E( ∂2L
∂α2∂α1

) E( ∂2L
∂α2∂θ2

) E( ∂
2L

∂α2
2 )

 .

=

I11 I12 I13 I14
I21 I22 I23 I24
I31 I32 I33 I34
I41 I42 I43 I44

 .

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



42 M. MOHIE EL-DIN, A. SADEK, SHAIMAA ELMEGHAWRY

Where:

I13 = I31 = 0; I14 = I41 = 0,

I23 = I32 = 0; I24 = I42 = 0,

I11 = − n
θ1

2

−
n∑
i=1

xi
2

(α1 + θ1xi)2
,

I22 =
n∑
i=1

1

(α1 + θ1xi)2
− n

(α1 + 1)2
,

I33 = −m
θ2

2

−
m∑
j=1

yj
2

(α2 + θ2yj)2
,

I44 =
m∑
j=1

1

(α2 + θ2yj)2
− m

(α2 + 1)2
,

I12 = I21 =
n∑
i=1

xi
(α1 + θ1xi)2

,

and

I34 = I43 = −
m∑
j=1

yj
(α2 + θ2yj)2

.

Using the Central limit theorem, we obtain the following theorem :

Theorem 1: As n→∞, m→∞; then

(
√
n(θ̂1 − θ1),

√
n(α̂1 − α1),

√
m(θ̂2 − θ2),

√
m(α̂2 − α2))

d→ N(0, I−1(Θ)).

Where d→ means converge in distribution, and I−1(Θ) is the inverse of the matrix I(Θ).

In order to establish the asymptotic normality of R, we first define:

d(Θ) = (
∂R

∂θ1
,
∂R

∂α1

,
∂R

∂θ2
,
∂R

∂α2

)T = (d1, d2, d3, d4)
T ,

where T is transpose operation, and

d1 = −θ1(2α1(α2 + 1)(θ1 + θ2) + α1θ2 + (α2 + 1)(θ1 + θ2) + α2θ1 + θ1 + 2θ2)

(α1 + 1)(α2 + 1)(θ1 + θ2)3

+
3θ1 (α1(α2 + 1)(θ1 + θ2)

2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2)

(α1 + 1)(α2 + 1)(θ1 + θ2)4

−α1(α2 + 1)(θ1 + θ2)
2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2

(α1 + 1)(α2 + 1)(θ1 + θ2)3
,

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



ESTIMATION OF STRESS-STRENGTH RELIABILITY 43

d2 =
θ1 (α1(α2 + 1)(θ1 + θ2)

2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2)

(α1 + 1)2(α2 + 1)(θ1 + θ2)3

−θ1 ((α2 + 1)(θ1 + θ2)
2 + θ2(θ1 + θ2))

(α1 + 1)(α2 + 1)(θ1 + θ2)3
,

d3 =
3θ1 (α1(α2 + 1)(θ1 + θ2)

2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2)

(α1 + 1)(α2 + 1)(θ1 + θ2)4

−θ1(2α1(α2 + 1)(θ1 + θ2) + α1(θ1 + θ2) + α1θ2 + α2θ1 + 3θ1)

(α1 + 1)(α2 + 1)(θ1 + θ2)3
,

and

d4 =
θ1 (α1(α2 + 1)(θ1 + θ2)

2 + (θ1 + θ2)(α1θ2 + α2θ1 + θ1) + 2θ1θ2)

(α1 + 1)(α2 + 1)2(θ1 + θ2)3

−θ1 (α1(θ1 + θ2)
2 + θ1(θ1 + θ2))

(α1 + 1)(α2 + 1)(θ1 + θ2)3
.

Hence; using Theorem 1, the asymptotic distribution of R̂, the MLE of R is defined as
√
n+m(R̂−R)

d→ N(0, B),

where

B = V ar(R̂)

= dT (Θ)I−1(Θ)d(Θ).
(4.6)

Therefore, using Eq.(4.6), an asymptotic 100(1− α)% confidence interval for R can be
obtained as:

R̂± Zα
2

√
V ar(R̂),

where Zα
2

is the upper α
2

precentile of the standard normal distribution.

5. BAYESIAN ESTIMATION

In this section, we provide the Bayes estimate of R where θ1, θ2, α1, α2 are unknown
parameters and all of these parameters having independent gamma prior distributions as
following:

π(θ1) ∼ Gamma(a1, b1),

π(θ2) ∼ Gamma(a2, b2),

π(α1) ∼ Gamma(a3, b3),

and
π(α2) ∼ Gamma(a4, b4).

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



44 M. MOHIE EL-DIN, A. SADEK, SHAIMAA ELMEGHAWRY

The joint posterior PDF is defined as

g(θ1, θ2, α1, α2/data) =
L(x, y/θ1, θ2, α1, α2)π(θ1)π(θ2)π(α1)π(α2)∫∞

0

∫∞
0

∫∞
0

∫∞
0
L(x, y/θ1, θ2, α1, α2)π(θ1)π(θ2)π(α1)π(α2)dθ1dθ2dα1dα2

.

Then

g(θ1, θ2, α1, α2/data) ∝ θ1
n

(α1 + 1)n
θ2
m

(α2 + 1)m

n∏
i=1

(α1 + θ1xi)
m∏
j=1

(α2 + θ2yj) e
−θ1

∑n
i=1 xi

× e−θ2
∑m
j=1 yj θ1

a1−1e−b1θ1θ2
a2−1e−b2θ2α1

a3−1e−b3α1α2
a4−1e−b4α2 .

(5.7)

5.1. Bayes estimators under Symmetric and Asymmetric loss function:
The Bayes estimate of reliability formula depending on the choice of the loss function. Two
different loss functions are used, symmetric and asymmetric loss function. If the amount
of loss assigned by a loss function to a positive error is equal to the negative error of the
same magnitude, then the loss function is called a symmetric loss function. In most of the
studies on estimation and prediction problems, authors prefer to use the squared error loss
function which is symmetric in nature. However, the use of the squared error loss function
is not appropriate particularly in these cases, where the losses are not symmetric. Thus in
order to make the statistical inferences more practical and applicable, we often needs to
choose an asymmetric loss function. A number of asymmetric loss functions proposed for
use, first is the linex loss function which suggested by Varian [20], and studied by several
others including Basu and Ebrahimi [3], Zellner [21]. Second is the general entropy loss
function which introduced by Calabria and Pulcini [5]. These asymmetric loss functions are
also studied by Braess and Dette [4], Pandey and Rao [14], Parsian and Kirmani [15], Sanku
Dey [16] and Soliman [19], who have used these loss function in different estimation and
prediction problem.
Then the equation of the Bayes estimate of the reliability is depend on the loss function, here
is the general equation for each type:
-The Bayes estimate under the squared error loss function, which is the posterior mean of R,
is given by:

R̂Se =

∫ ∞
0

∫ ∞
0

∫ ∞
0

∫ ∞
0

R g(θ1, θ2, α1, α2/data)dθ1dθ2dα1dα2.

-The Bayes estimate under the linex loss function is given by:

R̂Lx =
−1

c
ln

[ ∫ ∞
0

∫ ∞
0

∫ ∞
0

∫ ∞
0

e−cR g(θ1, θ2, α1, α2/data)dθ1dθ2dα1dα2

]
,

where c is constant, c > 0, see [21].
-The Bayes estimate under the general entropy loss function is given by:

R̂Ge =

[ ∫ ∞
0

∫ ∞
0

∫ ∞
0

∫ ∞
0

R−q g(θ1, θ2, α1, α2/data)dθ1dθ2dα1dα2

]−1/q
,

where q is constant, q > 0, see [5].
It is impossible to compute these integrals analytically. Two approaches can be used to
approximate these integrals, namely, Importance Sampling technique and MCMC technique.

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



ESTIMATION OF STRESS-STRENGTH RELIABILITY 45

5.2. Importance Sampling Technique
Importance Sampling Technique has suggested by Chen and Shao [6]. In statistics,
importance sampling is the name for the general technique of determining the properties of a
distribution by drawing samples from another distribution. The focus of importance sampling
here is to determine as easily and accurately as possible the properties of the posterior from
a representative sample from the second distribution.

Using Importance Sampling Technique, Eq.(5.7) can be written as

g(θ1, θ2, α1, α2/data) ∝ g1(θ1/data)g2(θ2/data)g3(α1/data)g4(α2/data)h(θ1, θ2, α1, α2/data),

where:

g1(θ1/data) ∝ Gamma(n+ a1, b1 +
n∑
i=1

xi),

g2(θ2/data) ∝ Gamma(m+ a2, b2 +
m∑
j=1

yj),

g3(α1/data) ∝ Gamma(a3, b3),

g4(α2/data) ∝ Gamma(a4, b4),

and

h(θ1, θ2, α1, α2/data) =

∏n
i=1 (α1 + θ1xi)

∏m
j=1 (α2 + θ2yj)

(α1 + 1)n (α2 + 1)m
.

As shown, all the above functions from g1(θ1/data) to g4(α2/data) follow gamma
distributions with different parameters, so it is quite simple to generate QLD parameters from
them. Assuming that a1, · · · , a4 and b1, · · · , b4 are known, and assuming initial values for
θ1, θ2, α1, α2. we can use the following Importance Sampling Algorithm:

• Step1: Generate θ11 from g1(θ1/data).
• Step2: Generate θ21 from g2(θ2/data).
• Step3: Generate α11 from g3(α1/data).
• Step4: Generate α21 from g4(α2/data).
• Step5: Repeat steps from 1 to 4, N times to obtain the vector

(θ11, θ21, α11, α21), · · · , (θ1N , θ2N , α1N , α2N).

Then
-An approximate Bayes estimate of R under squared error loss function can be obtained as

R̃impSe =

∑N
i=1Ri h(θ1i, θ2i, α1i, α2i/data)∑N
i=1 h(θ1i, θ2i, α1i, α2i/data)

,

- An approximate Bayes estimate of R under linex loss function can be obtained as

R̃impLx =
−1

c
log [

∑N
i=1 e

−cRi h(θ1i, θ2i, α1i, α2i/data)∑N
i=1 h(θ1i, θ2i, α1i, α2i/data)

],

-An approximate Bayes estimate of R under general entropy loss function can be obtained as:

R̃impGe = [

∑N
i=1Ri

−q h(θ1i, θ2i, α1i, α2i/data)∑N
i=1 h(θ1i, θ2i, α1i, α2i/data)

]−1/q,

where Ri = R(θ1i, θ2i, α1i, α2i). as defined in Eq.(2.3), for i = 1, · · · , N .

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



46 M. MOHIE EL-DIN, A. SADEK, SHAIMAA ELMEGHAWRY

5.3. MCMC Technique
The most general MCMC algorithm is the Metropolis-Hastings (MH) algorithm, which was
originally introduced by Metropolis et al. [13], and Hastings [8]. The Metropolis-Hastings
(MH) algorithm simulates samples from a probability distribution by making use of the full
joint density function and (independent) proposal distributions for each of the variables of
interest.
The joint posterior density function of θ1, θ2, α1, and α2 is given in Eq.(5.7). It is easily seen
that the posterior density functions of θ1, θ2, α1, and α2 are, respectively:

π1(θ1/data) ∝ Gamma

(
n+ a1, b1 +

n∑
i=1

xi

)
, (5.8)

π2(θ2/data) ∝ Gamma

(
m+ a2, b2 +

m∑
j=1

yj

)
, (5.9)

π3(α1/θ1, data) ∝ α1
a3−1e−b3α1

∏n
i=1 (α1 + θ1xi)

(α1 + 1)n
,

(5.10)

and

π4(α2/θ2, data) ∝
α2

a4−1e−b4α2
∏m

j=1 (α2 + θ2yj)

(α2 + 1)m
.

(5.11)

Therefore, easily samples of θ1 and θ2 can be generated by using Gamma distribution as
shown in Eqs.(5.8), and (5.9) respectively. However, the posterior distribution of α1 , α2

cannot be generated from a well known distributions. The Metropolis-Hastings algorithm,
can be used to solve this problem, as shown in the following algorithm.

• Step1: Start with initial value of α1,α2 such that α1
(0) = α̂1, and α2

(0) = α̂2.
• Step2: Set i = 1.
• Step3: Generate θ1(i) from π1(θ1/data).

• Step4: Generate θ2(i) from π2(θ2/data).
• Step5: Generate α1

(i) from π3(α1/θ1, data) using the Metropolis-Hastings algorithm
with the proposal distribution q1 as following:

– Generate α1
(∗) from the proposal distribution q1 = N(α1

(i−1), V ar(α1
(i−1))).

– Calculate the acceptance probability
r1(α1

(i−1), α1
(∗)) = Min[1, π3(α1

(∗)/θ1
(i),data)

π3(α1
(i−1)θ1

(i),data)
].

– Generate U from Uniform(0, 1).

– If U ≤ r1(α1
(i−1), α1

(∗)), accept the proposal distribution and set α1
(i) = α1

(∗) ,
otherwise set α1

(i) = α1
(i−1).

• Step6: Generate α2
(i) from π4(α2/θ2, data) using the Metropolis-Hastings algorithm

with the proposal distribution q2 as following:

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



ESTIMATION OF STRESS-STRENGTH RELIABILITY 47

– Generate α2
(∗) from the proposal distribution q2 = N(α2

(i−1), V ar(α2
(i−1))).

– Calculate the acceptance probability
r2(α2

(i−1), α2
(∗)) = Min[1, π4(α2

(∗)/θ2
(i),data)

π4(α2
(i−1)θ2

(i),data)
].

– Generate U from Uniform(0, 1).

– If U ≤ r2(α2
(i−1), α2

(∗)), accept the proposal distribution and set α2
(i) = α2

(∗) ,
otherwise set α2

(i) = α2
(i−1).

• Step7: Compute R(i) at (θ1
(i), θ2

(i), α1
(i), α2

(i)) using Eq.(2.3).
• Step8: Set i = i+ 1.
• Step9: Repeat steps from (3− 8) N times.

Then;
-An approximate Bayes estimate of R under squared error loss function is given as:

R̃MHSe
=

1

N −M

N∑
i=M+1

R(i).

-An approximate Bayes estimate of R under linex loss function is given as:

R̃MHLx
=
−1

c
log

[
1

N −M

N∑
i=M+1

e−cR
(i)

]
.

-An approximate Bayes estimate of R under general entropy loss function is given as:

R̃MHGe
=

[
1

N −M

N∑
i=M+1

(
R(i)

)−q]−1/q
.

where M is the burn-in units, N is the MCMC samples.

6. SIMULATION STUDY

In this section, we mainly present some simulation experiments to see the
performance of the mentioned methods for different sample sizes, (n,m) =
(10, 10), (20, 20), (30, 30), (50, 50), (70, 70), (100, 100). we simulated 1000 complete
samples from quasi lindely distribution with the parameter values; θ1 = 0.2, θ2 = 1.5, α1 =
2, α2 = 0.8 with true reliability value is 0.87399. We also compute the 95% confidence
intervals of R based on the observed Fisher information matrix. We compared the
performances of the MLE and the Bayes estimates in terms of mean squared errors (MSE’s).
Also two different techniques of Bayesian estimation (Importance, MCMC) are compared
for different loss error functions. Bayesian estimation for different loss error functions was
proposed with many values of c, q such that; c1 = −3(Lx1), c2 = 5(Lx2), q1 = −3 (Ge1),
q2 = 5 (Ge2).
Bayesian estimation studied under the informative gamma priors. For choosing suitable
hyper-parameters, the experimenters can incorporate their prior guess in terms of location
and precision for the parameter of interest. The gamma distribution for the priors has
mean = a/b, and variance = a/b2. We assume a small value of prior variance (0.01), and
take the mean equal to the true value of the parameter of interest. For each parameter prior
we solve the two equations of the mean and the variance, we obtain the following values of

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



48 M. MOHIE EL-DIN, A. SADEK, SHAIMAA ELMEGHAWRY

hyper-parameters :
a1 = 4, a2 = 225, a3 = 400, a4 = 64, and b1 = 20, b2 = 150, b3 = 200, b4 = 80. We
also computed the Bayes estimates based on 11000 samples and discard the first 1000 values
as burn-in.
The maximum likelihood estimator and asymptotic confidence intervals of R for different
(n,m) are obtained in Table 6.1. Bayes estimates of R using different techniques under
different loss error functions are obtained in Table 6.2.
Therefore, from this study of the simulation results we observed that:

• The performance of the Bayes estimators is better than maximum likelihood for all
different sample sizes.
• Mean squared error(MSE’s) for all estimation methods decreased as sample size

increased.
• As sample size increased, the asymptotic confidence intervals for R are improving, and

their lengths are decreasing. That means the estimated reliability becomes in the most
accurate interval.
• When the sample size increased, both Bayesian and maximum likelihood results become

close to each other.
• For Bayes estimators, importance sampling technique gives less MSE’s values, so it is

better than MCMC technique for the same priors values, and same number of generated
samples.
• General entropy, and linex loss error functions gave less MSE’s at specified values

of c, q. As shown Lx2, Ge2 acheived the best results for MCMC, but for importance
sampling technique Lx1, Ge1 are the best methods .

Table 6.1. Average estimate (mean squared error) for MLE, and average confidence length of the simulated
95%confidence intervals of R. all MSE values are multiplied by 10−3

Estimator MLE C.I.L C.I.U C.I. length

(10,10) 0.866095 0.734952 0.997237 0.262
(3.2129)

(20,20) 0.878225 0.787222 0.969227 0.182
(1.1262)

(30,30) 0.874831 0.797586 0.952077 0.155
(0.7609)

(50,50) 0.875794 0.816741 0.93706 0.120
(0.459333)

(70,70) 0.875149 0.823703 0.926595 0.103
(0.3349)

(100,100) 0.874276 0.834959 0.919893 0.084
(0.1554)

7. REAL DATA ANALYSIS

In this section we present the analysis of real data, introduced by Singh et al. [18]. The data
represent the waiting times (in minutes) before customer service of two banks A and B,
respectively. The use of Lindley distribution for the waiting times (bank A) data has been
originally discussed by Lindley [12]. Since then, many authors have suggested the data under
different set-up for Lindley distribution. We are interested in estimating the stress-strength

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



ESTIMATION OF STRESS-STRENGTH RELIABILITY 49

Table 6.2. Average estimates(mean squared error ) of R for different Bayes estimators under different error loss
functions. all MSE values are multiplied by 10−3

Est. Importance Sampling MCMC Technique

(m,n) Se Lx1 Lx2 Ge1 Ge2 Se Lx1 Lx2 Ge1 Ge2

(10,10) 0.86713 0.86862 0.86457 0.86829 0.86346 0.88787 0.88908 0.88577 0.8888 0.88494
(0.7516) (0.7058) (0.8421) (0.7134) (0.8934) (0.7611) (0.7748) (0.7459) (0.77) (0.7477)

(20,20) 0.87063 0.87147 0.86919 0.87128 0.86861 0.89199 0.89268 0.89081 0.89251 0.89038
(0.4769) (0.4611) (0.5075) (0.4637) (0.5239) (0.6747) (0.692) (0.6479) (0.6870) (0.6403)

(30,30) 0.87073 0.87135 0.86967 0.87121 0.86925 0.89157 0.89208 0.89071 0.89196 0.8904
(0.3591) (0.3491) (0.3784) (0.3508) (0.3883) (0.5684) (0.5822) (0.5464) (0.5785) (0.5396)

(50,50) 0.87168 0.87206 0.87104 0.87198 0.87079 0.88997 0.89032 0.88938 0.89023 0.88916
(0.2536) (0.2486) (0.2626) (0.2496) (0.2670) (0.4366) (0.4458) (0.4216) (0.4433) (0.4167)

(70,70) 0.873434 0.873697 0.872995 0.873636 0.872827 0.88874 0.889014 0.88828 0.888946 0.888114
(0.2189) (0.217) (0.2226) (0.2173) (0.2244) (0.3755) (0.3823) (0.3645) (0.3805) (0.3609)

(100,100) 0.875249 0.875418 0.874965 0.875378 0.874857 0.885949 0.886165 0.885586 0.886112 0.885455
(0.1384) (0.1387) (0.138) (0.1386) (0.138) (0.2432) (0.2477) (0.2359) (0.2465) (0.2334)

parameter R = P (Y < X) where X and Y denotes the customer service time in Bank A
and B (Data set 1, 2) respectively. The data sets are presented below:

Data set 1: X (n=100)
0.8, 0.8, 1.3, 1.5, 1.8, 1.9, 1.9, 2.1, 2.6, 2.7, 2.9, 3.1, 3.2,3.3, 3.5, 3.6,4.0, 4.1, 4.2, 4.2, 4.3,
4.3, 4.4, 4.4, 4.6, 4.7, 4.7, 4.8, 4.9, 4.9, 5.0, 5.3, 5.5, 5.7, 5.7, 6.1, 6.2, 6.2, 6.2, 6.3, 6.7, 6.9,
7.1, 7.1, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.2, 8.6, 8.6, 8.6, 8.8, 8.8, 8.9, 8.9, 9.5, 9.6, 9.7, 9.8, 10.7,
10.9, 11.0, 11.0, 11.1, 11.2, 11.2, 11.5, 11.9, 12.4, 12.5, 12.9, 13.0, 13.1, 13.3, 13.6, 13.7,
13.9, 14.1, 15.4, 15.4, 17.3, 17.3, 18.1, 18.2, 18.4, 18.9, 19.0, 19.9, 20.6, 21.3, 21.4, 21.9,
23.0, 27.0, 31.6, 33.1, 38.5.

Data set 2: Y (m=60)
0.1, 0.2, 0.3, 0.7, 0.9, 1.1, 1.2, 1.8, 1.9, 2.0, 2.2, 2.3, 2.3, 2.3, 2.5, 2.6, 2.7, 2.7, 2.9, 3.1, 3.1,
3.2, 3.4, 3.4, 3.5, 3.9, 4.0, 4.2, 4.5, 4.7, 5.3, 5.6, 5.6, 6.2, 6.3, 6.6, 6.8, 7.3, 7.5, 7.7, 7.7, 8.0,
8.0, 8.5, 8.5, 8.7, 9.5, 10.7, 10.9, 11.0, 12.1, 12.3, 12.8, 12.9, 13.2, 13.7, 14.5, 16.0, 16.5, 28.0.

First,we checked the suitability of Quasi Lindley distribution for the considered real data
sets. We, therefore, have provided the Kolmogorov-Smirnov (K-S),Anderson-Darling(A-D)
and Cramér-von Mises Statistics to test the goodness-of-fit of above data sets to the Quasi
Lindley distribution. The fitting summary has been presented in Table 7.1, which indicates
that the QLD fits well to Data Set 1 and Data Set 2.

Table 7.3. P-value (Statistic) of different goodness-of-fit tests for data set 1, 2.

K-S A-D Cramér-von

data set 1. 0.0654 (0.1290) 0.0217 (3.2033) 0.0501 (0.4610)

data set 2. 0.9287 (0.0677) 0.965 (0.2597) 0.9310 (0.0404)

Based on the MLEs θ̂1, α̂1, θ̂2, α̂2 the point estimate of R is 0.59 and the 95% confidence
interval of R is (0.25, 0.93). For real data sets, the maximum likelihood and Bayes estimates
of the stress-strength parameters and reliability are summarized in Table 7.2.

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



50 M. MOHIE EL-DIN, A. SADEK, SHAIMAA ELMEGHAWRY

Table 7.4. The MLEs and Bayes estimates of stress-strength parameters and reliability R from real data sets

Est. θ̂1 θ̂1 α̂1 α̂2 R̂

MLE 0.1 0.27 84.09 0.41 0.59

BayesIMP 0.13 0.59 1.44 0.66 0.81

BayesMH 0.1 0.54 22.13 0.69 0.77

8. CONCLUSION

In this paper, maximum likelihood and Bayesian estimation methods for stress-strength
reliability R were discussed, when X and Y both follow a quasi lindley distribution with
different parameters. We obtained the 95% confidence intervals of R based on the observed
Fisher information matrix. We proposed the Bayesian estimation based on independent
gamma priors under different error loss functions(squared, linex, and general entropy). we
suggested the Importance sampling, and MCMC techniques to generate samples from the
posterior distributions and then compute the Bayes estimates. Simulation study has been
introduced to investigate the performance and compare among all mentioned methods.
Simulation results suggest that the performance of the Bayes estimator is better than
maximum likelihood for all different sample sizes. Also, maximum likelihood method
provides very satisfactory results as the sample size increased.

REFERENCES

1. Al-Mutairi, D.K., Ghitany, M.E. & Kundu, D. (2013). Inferences on stress-strength
reliability from Lindley distributions. Communications in Statistics-Theory and
Methods, 42, 1443-1463.

2. Al-Mutairi, D.K., Ghitany & M.E.,Kundu, D. (2015). Inferences on stress-strength
reliability from weigthed Lindley distributions. Communications in Statistics-Theory
and Methods, 44, 4096-4113.

3. Basu, A.P & Ebrahimi, N.(1991). Bayesian Approach to life Testing and Reliability
Estimation Using Asymmetric Loss Function. Jour. Stat. Plann. Infer, 29, 21-31.

4. Braess, D. & Dette, H.(2004). The asymptotic minimax risk for the estimation of
constrained binomial and multinomial probabilities. Sankhyaind.J.statist.66(4), 707-
732.

5. Calabria, R. & Pulcini, G. (1994). An engineering approach to Bayes estimation for the
Weibull distribution. Microelectronic Reliability, 34 (5),789-802.

6. Chen, M.H. & Shao, Q.M. (1999). Monte Carlo estimation of Bayesian credible and
HPD intervals. Journal of Computational and Graphical Statistics, 8(1), 69-92.

7. Church, J.D. & Harris, B. (1970). The estimation of reliability from stress-strength
relationships. Technometrics, 12, 49-54.

8. Hastings, WK. (1970). Monte Carlo sampling methods using Markov chains and their
applications. Biometrika, 57, 97-109.

9. Khan, Adil H. & Jan, T. R. (2015). Estimation of stress-strength reliability model using
finite mixture of two parameter Lindley distributions. Journal of Statistics Applications
and Probability, 4(1), 147-159.

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)



ESTIMATION OF STRESS-STRENGTH RELIABILITY 51

10. Kotz, S. , Lumelskii,Y. & Penskey,M. (2003). The stress–strength model and its
generalizations and applications. World Scientific, Singapore.

11. Krishnamoorthy, K., Mukherjee, S. & Guo, H. (2007). Inference on reliability in two-
parameter exponential stress-strength model. Metrika, 65, 261-273.

12. Lindley, D.V.(1958) Fiducial distributions and bayes theorem. Journal of the Royal
Statistical Society, 20, 102-107.

13. Metropolis, N., Rosenbluth, AW., Rosenbluth, MN., Teller, AH. & Teller, E., (1953).
Equations of state calculations by fast computing machine. Journal of Chemical
Physics, 21, 1087-1092.

14. Pandey, H. & Rao, A. K. (2009). Bayesian estimation of the shape parameter of a
generalized Pareto distribution under asymmetric loss functions. Hacettepe Journal of
Mathematics and Statistics, 38(1), 69-83.

15. Parsian, A. & Kirmani, S. N. U. A. (2002). Estimation under LINEX loss function.
Hand book of applied econometrics and statistical inference, Statistics Textbook and
Monograph.165. New York: Marcel Dekker Inc, 53-76.

16. Dey, S. (2010). Bayesian estimation of the shape parameter of the generalized
exponential distribution under different loss functions. Pak. J. Statist. Oper. Res, 62,
163-174.

17. Shanker, R., Sharma, S. & Shanker, R. (2013). A quasi Lindley distribution. African
Journal of Mathematics and Computer Science Research, 6(4), 64-71.

18. Singh, S. K., Singh, U., & Sharma, V. K. (2014). Estimation on system reliability
in generalized Lindley stress-strength model.Journal of Statistics Applications and
Probability, 3(1), 61-75.

19. Soliman, A.A. (2000). Comparison of Linex and Quadratic Bayes estimators for the
Rayleigh Distribution. Commun.Statis.Theory Meth, 29(1), 95-107.

20. Varian, H.R.(1975). A Bayesian Approach to Reliability Real Estate Assessment.
Amsterdam, North Holland, 195-208.

21. Zellner, A. (1986). Bayesian estimation and prediction using asymmetric loss functions.
Journal of American Statistical Associations, 81(394), 446-451.

Copyright c© 2018 ASSA. Adv Syst Sci Appl (2018)