152

CHAPTER – 5

STRESS-STRENGTH RELIABILITY ESTIMATION

5.1 Introduction

There are appliances (every physical component possess an inherent

strength) which survive due to their strength. These appliances receive a

certain level of stress and sustain. But if a higher level of stress is applied

then their strength is unable to sustain and they break down. Suppose Y

represents the ‘stress’ which is applied to a certain appliance and X

represents the ‘strength’ to sustain the stress, then the stress-strength

reliability is denoted by R= P(Y<X), if X,Y are assumed to be random.

The term stress is defined as: It is failure inducing variable. It is

defined as stress (load) which tends to produce a failure of a component or

of a device of a material. The term load may be defined as mechanical load,

environment, temperature and electric current etc.

The term strength is defined as: The ability of component, a device or

a material to accomplish its required function (mission) satisfactorily

without failure when subjected to the external loading and environment.

Therefore strength is failure resisting variable.

Stress-strength model: The variation in ‘stress’ and ‘strength’ results

in a statistical distribution and natural scatter occurs in these variables when

the two distributions interfere. When interference becomes higher than

strength, failure results. In other words, when probability density functions

of both stress and strength are known, the component reliability may be

determined analytically.

Therefore R= P(Y<X) is reliability parameter R. Thus R= P(Y<X) is

the characteristic of the distribution of X and Y.

153

Let X and Y be two random variables such that X represents

“strength” and Y represents “stress” and X,Y follow a joint pdf f(x,y),

then reliability of the component is:

( ) f(x,y)dy dxx

R P Y X+∞

−∞ −∞

= < = ∫ ∫ (5.1.1)

where P(Y<X) is a relationship which represents the probability, that the

strength exceeds the stress and f(x,y) is joint pdf of X and Y.

If the random variables are statistically independent, then

f(x,y) = f(x) g(y) so that

f(x)g(y)dy dxx

R∞

−∞ −∞

= ∫ ∫ (5.1.2)

∫∞−

=x

y )(G g(y)dyx

yR G (x)f(x)dx∞

−∞

= ∫ (5.1.3)

where f(x) and g(y) are pdf’s of X and Y respectively.

The concept of stress-strength in engineering devices has been one

of the deciding factors of failure of the devices. It has been customary to

define safety factors for longer lives of systems in terms of the inherent

strength they have and the external stress being experienced by the systems.

If Xo is the fixed strength and Yo is the fixed stress, that a system is

experiencing, then the ratio Yo oX is called safety factor and the difference

Xo - Yo is called safety margin. Thus in the deterministic stress-strength

154

situation the system survives only if the safety factor is greater than 1 or

equivalently safety margin is positive.

In the traditional approach to design of a system the safety factor or

margin is made large enough to compensate uncertainties in the values of

the stress and strength variates. Thus uncertainties in the stress and strength

of a system make the system to be viewed as random variables. In

engineering situations the calculations are often deterministic. However, the

probabilistic analysis demands the use of random variables for the concepts

of stress and strength for the evaluation of survival probabilities of such

systems. This analysis is particularly useful in situations in which no fixed

bound can be put on the stress. For example, with earth quakes, floods and

other natural phenomena, the shortcomings may result in failures of systems

with unusually small strengths. Similarly when economics rather than

safety is the primary criterion, the comparison of survival performance can

be better studied by knowing the increase in failure probability as the stress

and strength approach one another. In foregoing lines it is mentioned that

the stress-strength variates are more reasonable to be random variables than

purely deterministic.

Let the random samples 1 2, , ,..., kY X X X being independent, G(y) be

the continuous cdf of Y and F(x) be the common continuous cdf

of 1 2, ,..., kX X X . The reliability in a multicomponent stress- strength model

developed by Bhattacharyya and Johnson (1974) is given by

s,k 1 2 kR [ s (X ,X ,...,X ) Y]P at least of the exceed=

( ) ∫∑∞

∞−=−= dF(y),G(y)G(y)k i-ki

k

sii ][]1[ (5.1.4)

155

where 1 2, ,..., kX X X are iid with common cdf F(x), this system is subjected

to common random stress Y.

The R.H.S of the equation (5.1.3) is called the survival probability of

a system of single component having a random strength X, and

experiencing a random stress Y. Let a system consist of say k components

whose strengths are given by independently identically distributed random

variables with cumulative distribution function F(.) each experiencing a

random stress governed by a random variable ‘Y’ with cumulative

distribution function G(.).The above probability given by (5.1.4) is called

reliability in a multi-component stress-strength model ( Battacharya and

Johnson, 1974).

A salient feature of probability mentioned in (5.1.1) is that the

probability density functions of strength and stress variates will have a non-

empty overlapping ranges for the random variables. In some cases the

failure probability may strongly depend on the lower tail of the strength

distribution.

When the stress or strength is not determined by either sum or the

product of many small components, it may be the extremes of these small

components that decide the stress or the strength. Extreme ordered variates

have proved to be very useful in the analysis of reliability problems of this

nature. Suppose that a number of random stresses say

1 2 3 m(Y ,Y ,Y ,.......,Y ) with cumulative distribution function G(.) are acting on

a system whose strengths are given by ‘m’ random variables

1 2 3 n(X ,X ,X ,.......,X ) with cumulative distribution function F(.). If ‘V’ is the

maximum of 1 2 3 mY ,Y ,Y ,.......,Y and ‘U’ is the minimum of

1 2 3 nX ,X ,X ,.......,X , the survival of the system depends on whether or not

‘U’ exceeds ‘V’. That is the survival probability of such a system is

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explained with the help of distributions of extreme order statistics in

random samples of finite sizes.

Thus the above introductory lines indicate that whether it is through

extreme values or independent variates like Y, X in a single component or

multicomponent system with stress and strength factors, the reliability in

any situation ultimately turns out to be a parametric function of the

parameters in probability distribution of the concerned variates. If some or

all of the parameters are not known, evaluation of the survival probabilities

of a stress-strength model leads to estimation of a parametric function.

As mentioned in the introduction, several authors have taken up the

problem of estimating survival probability in stress-strength relationship

assuming various lifetime distributions for the stress-strength random

variates. Some recent works in this direction are Kantam et al. (2000),

Kantam and Srinivasa Rao (2007), Srinivasa Rao et al. (2010b) and the

references therein.

In this chapter, we present estimation of R using maximum likelihood

(ML) estimates and moment (MOM) estimates of the parameters in Section

5.2. Asymptotic distribution and confidence intervals for R are given in

Section 5.3. Simulation studies are carried out in Section 5.4 to investigate

bias and mean squared errors (MSE) of the MLE and MOM of R as well as

the lengths of the confidence interval for R. The conclusions and comments

are provided in Section 5.5.

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5.2 Estimation of stress-strength reliability

5.2.1 Estimation of stress-strength reliability using ML and moment estimates :

The purpose of this section is to study the inference of R P(Y X)= < ,

where 1~ ( )X IRD σ , 2~ ( )Y IRD σ and they are independently distributed.

The estimation of reliability is very common in statistical literature. The

problem arises in the context of reliability of a component of strength, X,

subject to a stress Y. Thus R P(Y X)= < is a measure of reliability of the

system. The system fails if and only if the applied stress is greater than its

strength. We obtain the maximum likelihood estimator (MLE) and moment

estimator (MOM) of R and obtain its asymptotic distribution. The

asymptotic distribution has been used to construct an asymptotic confidence

interval. The bootstrap confidence interval of R is also proposed. Assuming

that the different scale parameters are not known, we obtain the MLE and

MOM of R.

Suppose X and Y are random variables independently distributed as

1~ ( )X IRD σ and 2~ ( )Y IRD σ . Therefore,

dydxyx

X)P(YR yxx

22

21

e2

e2

3

22

0 03

21

σ−

σ−∞ σσ=<= ∫ ∫

dxx

xx

22

21

ee2

03

21

σ−

σ−∞

∫σ=

22

21

21

σ+σσ= . (5.2.1)

Note that, 2

1

σσ

,1

=+

= λλλ

R 2

2

(5.2.2)

158

Therefore, .01(

2 >+

=∂∂

22 )λλ

λR

R is an increasing function in λ.

Now to compute R, we need to estimate the parameters 21 and σσ ,

and these are estimated by using MLE and MOM.

5.2.2 Method of Maximum Likelihood Estimation (MLE)

Suppose 1 2 3 nX ,X ,X ,...,X is a random sample from inverse Rayleigh

distribution (IRD) with 1σ and 1 2 3 mY ,Y ,Y ,...,Y is a random sample from

IRD with 2σ . The log-likelihood function of the observed samples is

2 21 2 1 2 1 2

1 1 1 1

1 1( , ) ( ) log2 2 log 2 log log log

n m n m3 31 j2 2

i j i ji j

Log L m n n m x yx y= = = =

σ σ = + + σ + σ − σ − σ − −∑ ∑ ∑ ∑

(5.2.3)

The MLEs of 2121 ˆandˆsay,and σσσσ respectively can be obtained as

,1

ˆ

2i

1

∑=σ

x

n (5.2.4)

∑=σ

2j

2 1ˆ

y

m. (5.2.5)

The MLE of stress-strength reliability R becomes

.ˆˆ

ˆˆ22

21

21

1 σ+σσ=R (5.2.6)

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5.2.3 Method of Moment (MOM) Estimation

If y,x are the sample means of samples on strength and stress

variates respectively, then moment estimators (MOM) of 1 2σ ,σ are

1σ x / π= and 2σ y / π= respectively. Estimate of R using MOM method

adopting invariance property for moment estimation, we get

.~~

~ˆ

22

21

2

12 σ+σ

σ=R (5.2.7)

5.2.4 Asymptotic distribution and confidence intervals

In this Section, first we obtain the asymptotic distribution of

)ˆ,ˆ(ˆ21 σσθ = and then we derive the asymptotic distribution of 1R̂ . Based on

the asymptotic distribution of 1R̂ , we obtain the asymptotic confidence

interval of R.

Let us denote the Fisher’s information matrix of ),( 21 σσθ = as

( ).2,1);()( == ji,θIθI ij Therefore,

=

2221

1211

II

IIθI )( . (5.2.8)

where

2

11 2 21 1

log 4L nI E

∂= − = ∂ σ σ.

2

22 2 22 2

log 4L mI E

∂= − = ∂ σ σ.

160

2

12 211 2

log0

LI I E

∂= = − = ∂ ∂ σ σ.

As , ,n m→ ∞ → ∞

( )11 1 2 2 1 2ˆ ˆ( ), ( ) 0, ( , ) ,dn m N A− − − → σ σ σ σ σ σ

where 111 2

22

0( , )

0

aA

a

=

σ σ ,

and 111 11 2

1

4na I= =

σ,

122 22 2

2

4.ma I= =

σ

To obtain asymptotic confidence interval for R, we proceed as follows

(Rao, 1973) :

21 2

1 1 2 2 2 21 1 2

2( , )

( )

Rd

σ σσ σσ σ σ

∂= =∂ +

21 2

2 1 2 2 2 22 1 2

2( , )

( )

Rd

∂ −= =∂ +

σ σσ σσ σ σ

.

This gives

2 21 1 1 1 2 2 2 1 2

ˆ ˆ ˆ( ) var( ) ( , ) var( ) ( , )Var R d d= +σ σ σ σ σ σ

2 2

2 21 21 1 2 2 1 2( , ) ( , )

4 4d d

n m= +σ σσ σ σ σ

2 22 2 2 2

1 1 2 2 1 22 2 2 2 2 21 2 1 2

2 2

4 4( ) ( )n m

−= + + +

σ σ σ σ σ σσ σ σ σ

161

[ ]2 1 1(1 )R R

n m = − +

(5.2.9)

Thus we have the following result,

As , ,n m→ ∞ → ∞

1ˆ

(0,1)1 1

(1 )

dR RN

R Rn m

− → − +

.

Hence the asymptotic 100(1 )%−α confidence interval for R would be

( )1 1,L U , where

( )1 1 (1 / 2) 1 11 1ˆ ˆ ˆ1L R Z R Rn m−

= − − +

α , (5.2.10)

and

( )1 1 (1 / 2) 1 11 1ˆ ˆ ˆ1 .U R Z R Rn m−

= + − +

α (5.2.11)

Where (1 / 2)Z −α is the th(1 /2)−α percentile of the standard normal distribution

and 1R̂ is given by equation (5.2.6).

5.2.5 Exact confidence interval

Let 1 2 3 nX ,X ,X ,...,X and 1 2 3 mY ,Y ,Y ,...,Y are two independent

random samples of size n, m respectively, drawn from inverse Rayleigh

distribution with scale parameters 1 2andσ σ . Thus, 2 21 1

1 1and

n m

i ji jx y= =∑ ∑ are

independent gamma random variables with parameters 2 21 2( , )and( , )n mσ σ

162

respectively. We see that 2 21 22 2

1 1

1 12 and 2

n m

i ji jx y= =∑ ∑σ σ are two independent Chi-

square random variables with 2n and 2m degrees of freedom respectively.

Thus, 1R̂ in equation (5.2.6) could be rewritten as 12

21 2

1

ˆˆ 1 .ˆ

R

−

= +

σσ

Using (5.2.4) and (5.2.5) we obtain

12

21 2

1

ˆ 1R F

−

= +

σσ

(5.2.12)

where 2 2

1122

21

1

1

n

i im

j j

m xF

ny

=

=

=∑

∑σσ

is an F distributed random variable with (2n, 2m)

degrees of freedom. From equations (5.2.5) and (5.2.12), we see that F

could be written as 1

1

ˆ1ˆ 1

R RF

RR

− = − .

Using F as a pivotal quantity, we obtain a 100(1 )%−α confidence interval

for R as ( )2 2,L U , where

1

11

2 (1 / 2)ˆ1 ( 1) (2 ,2 )R

L F m n−

− = + − α (5.2.13)

1

11

2 ( / 2)ˆ1 ( 1) (2 ,2 )R

U F m n−

= + − α (5.2.14)

where ( /2) (1 /2)(2 ,2 ) and (2 ,2 )F m n F m nα α− are the lower and upper th( / 2)α

percentiles of F distribution with 2m and 2n degrees of freedom.

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5.2.6 Bootstrap confidence intervals

In this subsection, we propose to use confidence intervals based on

percentile bootstrap method (we call it from now on as Boot-p) based on the

idea of Efron (1982). We illustrate briefly how to estimate confidence

interval of R using this method.

Step 1: Draw random samples 1 2, ,..., nX X X and 1 2, ,..., mY Y Y from the

populations of X and Y respectively.

Step 2: Using the random samples 1 2, ,..., nX X X and 1 2, ,..., mY Y Y , generate

bootstrap samples 1 2, ,..., nx x x∗ ∗ ∗ and * * *1 2, ,..., my y y respectively. Compute the

bootstrap estimates of 1σ and 2σ , say 1∗σ and 2

∗σ respectively. Using 1∗σ

and 2∗σ and equation (5.2.6), compute the bootstrap estimate of R, say R̂∗ .

Step 3: Repeat step 2, NBOOT times, where N=1000.

Step 4: Let ˆ( ) ( )G x P R x∗= ≤ be the empirical distribution function of R̂∗ .

Let 3ˆ ( / 2)boot pL R∗

−= α and 3

ˆ (1 / 2)boot p

U R ∗

−

= −α be the 100 / 2α th and the

100(1 / 2)−α th empirical percentiles of the R̂∗ values respectively. The

small sample comparisons are studied through simulation in Section 5.4.

5.3 Estimation of reliability in multi-component stress-strength Model

Assuming that F(.) and G (.) are inverse Rayleigh distributions with

unknown scale parameters 1 2,σ σ and that independent random samples

1 2 nX X <....< X < and 1 2 mY < Y .... Y< < are available from F(.) and G(.)

respectively. The reliability in multi-component stress-strength for inverse

Rayleigh distribution using (5.1.4) is

164

( ) 2 2 22 2 1

2( / ) ( / ) ( / )1

, 30

2 [1 ] [ ]

ky y yi k i

s ki

i s

R K e e e dyy

σ σ σσ∞− − −−

== −∑ ∫

( ) 2 21

1/ 1/

0

[1 ] [ ]k

i k i

ii s

k t t dtλ λ −

== −∑ ∫ where

2 21

2/ 1

22

yt e andσ σλσ

−= =

( ) 2 2

1

2 1 1 /

0

[1 ][ ] ifk

i k i

i

i s

k z z dz z tλ λλ − + −

=

= − =∑ ∫

( ) 2 2 ( , 1).k

i

i s

k k i iλ β λ=

= − + +∑

After the simplification, we get

-1

2 2

0,

!( - ) , since and are integers.

( - )!

ik

i s js k

kR k j k i

k iλ λ

= =

= + ∑ ∏ (5.3.1)

The probability given in (5.3.1) is called reliability in a

multicomponent stress-strength model (Bhattacharyya and Johnson 1974).

Suppose a system with k identical components, functions, if (1 )s s k≤ ≤ or

more of the components simultaneously operate. In its operating

environment, the system is subjected to a stress Y which is a random

variable with cdf G(.). The strengths of the components, that is the

minimum stresses to cause failure are independent and identically

distributed random variables with cdf F(.). Then the system reliability,

which is the probability that the system does not fail is the function

,s kR given in (5.3.1).

If 1 2,σ σ are not known, it is necessary to estimate

1 2,σ σ to estimate

,s kR . In this section, we estimate 1 2,σ σ by ML method and method of

moment thus giving rise to two estimates. The estimates are substituted in

λ to get an estimate of ,s kR using equation (5.3.1). The theory of methods

of estimation is explained below.

It is well known that the method of maximum likelihood estimation

(MLE) has invariance property. When the method of estimation of

165

parameter is changed from ML to any other traditional method, this

invariance principle does not hold good to estimate the parametric function.

However, such an adoption of invariance property for other optimal

estimators of the parameters to estimate a parametric function is attempted

in different situations by different authors. Travadi and Ratani (1990),

Kantam and Srinivasa Rao (2002) and the references therein are a few such

instances. In this direction, we have proposed some estimators for the

reliability of multicomponent stress-strength model by considering the

estimators of the parameters of stress, strength distributions by standard

methods of estimation in inverse Rayleigh distribution. The MLEs of

1 2and aredenotedasσ σ 1σ and 2σ .

The asymptotic variance of the MLE is given by

12 2 2( log / ) / 4 ; 1,2i iE L n iσ σ−

− ∂ ∂ = = when m=n. (5.3.2)

The MLE of survival probability of multicomponent stress-strength model

is given by ,s kR with λ is replaced by 1 2/λ σ σ= in (5.3.1).

The second estimator, we propose here is ,s kR with λ is replaced

by 1 2/λ σ σ= in (5.3.1). Thus for a given pair of samples on stress, strength

variates we get two estimates of ,s kR by the above two different methods.

The asymptotic variance (AV) of an estimate of ,s kR which a function of

two independent statistics (say) 1 2t ,t is given by (Rao, 1973).

2 2

, ,

, 1 2

1 2

R RˆAV(R )=AV(t ) AV(t )s k s k

s k σ σ ∂ ∂+ ∂ ∂

(5.3.3)

Where 1 2t , t are to be taken in two different ways namely, exact ML and

method of moment estimators. Unfortunately, we can find the variance of

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inverse Rayleigh distribution, the asymptotic variance of ,s kR is obtained

using MLE only.

From the asymptotic optimum properties of MLEs (Kendall and

Stuart, 1979) and of linear unbiased estimators (David, 1981), we know that

MLEs are asymptotically equally efficient having the Cramer-Rao lower

bound as their asymptotic variance as given in (5.3.2). Thus from equation

(5.3.3), the asymptotic variance of ,R s k obtained when ( 1 2t ,t ) are replaced

by MLE.

To avoid the difficulty of derivation of ,R s k , we obtain the

derivatives of ,R s k for (s,k)=(1,3) and (2,4) separately, they are given by

2

1 , 3 1 , 3

2 2

2 2

1 22 2

6 6and

(3 ) (3 )

R Rλ λσ σσ λ σ λ

∂ −= =∂ ∂+ +

.

2 2

2 , 4 2 , 4

2 22 2 2 2

1 2

2 2

24 (2 7) 24 (2 7)and

(3 )(4 ) (3 )(4 )

R Rλ λ λ λσ σσ λ λ σ λ λ

∂ − + ∂ += =∂ ∂ + + + +

.

where 1 2/λ σ σ=

Thus 4

1 , 32 4

9 1 1ˆAV(R )=(3 ) n m

λλ

+ +

[ ]4 2 2

4

2 , 42 2

144 (2 7) 1 1ˆAV(R )=(3 )(4 ) n m

λ λλ λ

+ + + +

As n → ∞ , , ,

,

ˆN(0,1)

ˆAV(R )s k s k

d

s k

R R− → ,

and the asymptotic confidence - 95% confidence interval for ,s kR is given

by , ,ˆ ˆ1.96 AV(R )s k s kR ∓ .

167

The asymptotic confidence - 95% confidence interval for 1,3R is given by

2

1,3 2 2

3 1 1ˆ 1.96(3 )

Rn m

λλ

+ +

∓

The asymptotic confidence - 95% confidence interval for 2,4R is given by

2 2

2,4 22 2

12 (2 7) 1 1ˆ 1.96(3 )(4 )

Rn m

λ λ

λ λ

+ + + +

∓

The small sample comparisons are studied through simulation in Section

5.4.

5.4 Simulation study and Data analysis

5.4.1 Simulation study

In this section, we present some results based on Monte - Carlo

simulation, to compare the performance of the estimates of R and ,s kR using

ML and MOM estimators mainly for some sample sizes. We consider the

following some sample sizes; (n, m) = (5, 5), (10, 10), (15, 15), (20, 20) and

(25, 25). In both estimations we take 1 2( , )σ σ =(1, 3), (1, 2.5), (1, 2), (1,

1.5), (1, 1), (1.5, 1), (2.5, 1) and (3,1). All the results are based on 3000

replications. From each sample, we compute the estimates of 1 2( , )σ σ using

ML and Method of Moment estimation. Once we estimate 1 2( , )σ σ , we

obtain the estimates of R by substituting in (5.2.1) and (5.2.2) respectively.

Also obtain the estimates of ,s kR by substituting in 5.3.1. for (s,k)=(1,3) and

(2,4) respectively. We report the average biases of R in Table 5.4.1, ,s kR in

Table 5.4.5, mean squared errors (MSEs) of R in Table 5.4.2 and ,s kR in

Table 5.4.6 over 3000 replications. We also compute the 95% confidence

168

intervals based on the asymptotic distribution, exact distribution and Boot-p

method of 1R̂ . We report the average confidence lengths in Table 5.4.3 and

the coverage probabilities in Table 5.4.4 based on 1000 replications.

Average confidence length and coverage probability of the simulated 95%

confidence intervals of ,s kR are given in Table 5.4.7. Some of the points are

quite clear from this simulation. Even for small sample sizes, the

performance of the R using MLEs is quite satisfactory in terms of biases

and MSEs as compared with MOM estimates. When 1 2σ σ< , the bias is

positive and when 1 2σ σ> , the bias is negative. Also the absolute bias

decreases as sample size increases in both methods which is a natural trend.

It is observed that when m=n and m, n increase then MSEs decrease as

expected. It satisfies the consistency property of the MLE of R. As

expected, the MSE is symmetric with respect to 1σ and 2σ . For example, if

(n,m)=(10,10), MSE for 1 2( , )σ σ =(3,1) and MSE for 1 2( , )σ σ =(1,3) is the

same in case of MLE whereas MOM show approximately symmetric. The

length of the confidence interval is also symmetric with respect to

1 2( , )σ σ and decreases as the sample size increases. For (n, m)=(5, 5), we

find that average length of 2 2( , )L U < average length of 1 1( , )L U < average

length of 3 3( , )L U in all combinations of 1 2( , )σ σ . For other combinations of

(n,m), we find that average length of 2 2( , )L U < average length of 3 3( , )L U in

the case of 1 2σ σ≤ , whereas average length of 2 2( , )L U < average length of

1 1( , )L U < average length of 3 3( , )L U when 1 2σ σ> . Particularly, for a fixed

2σ when 1 2.0σ ≥ , the average length of exact confidence intervals and

Boot-p confidence intervals are almost same. Comparing the average

coverage probabilities, it is observed that for most of sample sizes, the

coverage probabilities of the confidence intervals based on the asymptotic

results reach the nominal level as compared with other two confidence

169

intervals. However, it is slightly less than 0.95. The performance of the

bootstrap confidence intervals is quite good for small samples. For all

combinations of (n,m) and 1 2( , )σ σ , the coverage probabilities in exact

confidence intervals is moderate and away from nominal level 0.95. The

overall performance of the confidence interval is quite good for asymptotic

confidence intervals. The simulation results also indicate that MLE shows

better performance than MOM in the average bias and average MSE for

different choices of the parameters. The exact confidence intervals are

preferable in the case of average short length of confidence intervals,

whereas asymptotic confidence intervals are advisable to use with respect to

coverage probabilities for different choices of the parameters.

Whereas the following points are observed for ,s kR for simulation

study. The true value of reliability in multicomponent stress-strength

increases when 1 2σ σ≤ otherwise true value of reliability decreases. Thus

the true value of reliability increases as λ decreases and vice-versa. Both

bias and MSE decreases as sample size increases for both methods of

estimation in reliability. With respect to bias, moment estimator is closer to

exact MLE in most of the parametric and sample combinations. Also the

bias is negative when 1 2σ σ≤ and in other cases bias is positive in both

situations of (s,k). With respect to MSE, MLE shows first preference than

method of moment estimation. The length of the confidence interval also

decreases as the sample size increases. The coverage probability is close to

the nominal value in all cases for MLE. The overall performance of the

confidence interval is quite good for MLE. The simulation results also show

that there is no considerable difference in the average bias and average

MSE for different choices of the parameters, whereas there is considerable

difference in MLE and MOM. The same phenomenon is observed for the

170

average lengths and coverage probabilities of the confidence intervals using

MLE.

5.4.2 Data analysis

We present a data analysis for two data sets reported by Lawless

(1982) and Proschan (1963). The first data set is obtained from Lawless

(1982) and it represents the number of revolution before failure of each of

23 ball bearings in the life tests and they are as follows:

Data Set I: 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96,

54.12, 55.56, 67.80, 68.44, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12,

105.84, 127.92, 128.04, 173.40.

Gupta and Kundu (2001) have fitted gamma, Weibull and Generalized

exponential distribution to this data.

The second data set is obtained from Proschan (1963) and represents

times between successive failures of 15 air conditioning (AC) equipment in

a Boeing 720 airplane and they are as follows:

Data Set II: 12, 21, 26, 27, 29, 29, 48, 57, 59, 70, 74, 153, 326, 386, 502.

We fit the inverse Rayleigh distribution to the two data sets

separately. We used the Kolmogorov-Smirnov (K-S) tests for each data set

to fit the inverse Rayleigh model. It is observed that for Data Sets I and II,

the K-S distances are 0.12091 and 0.21378 with the corresponding p values

are 0.85028 and 0.43879 respectively. For data sets I and II, the chi-square

values are 0.3052 and 2.6383 respectively. Therefore, it is clear that inverse

Rayleigh model fits quite well to both the data sets. We plot the empirical

survival functions and the fitted survival functions in Figures 5.4.1 and

5.4.2. Figures 5.4.1 and 5.4.2 show that the empirical and fitted models are

very close for each data set.

171

Based on estimates of 1σ and 2σ the MLE of R becomes 0.7043 and

the corresponding 95% confidence interval becomes (0.56882, 0.83978).

We also obtain the 95% Boot-p confidence intervals as (0.54543, 0.81669).

The MLE of ,s kR for (s,k)=(1,3) become 0.5573 and the corresponding

95% confidence interval becomes (0.39684, 0.71776). The MLE of ,s kR for

(s,k)=(2,4) become 0.3492 and the corresponding 95% confidence interval

becomes (0.16380, 0.53472).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 40 80 120 160 200 240

Empirical survival function

Fitted survival function

Figure 5.4.1: The empirical and fitted survival functions for Data Set I.

172

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300 350 400 450 500 550

Empirical survival function

Fitted survival function

Figure 5.4.2: The empirical and fitted survival functions for Data Set II.

5.5 Conclusions

We compare two methods of estimating ( )R P Y X= < when Y and

X both follow inverse Rayleigh distributions with different scale

parameters. We provide MLE and MOM procedure to estimate the

unknown scale parameters and use them to estimate of R. We also obtain

the asymptotic distribution of estimated R and that was used to compute the

asymptotic confidence intervals. The simulation results indicate that MLE

shows better performance than MOM in the average bias and average MSE

for different choices of the parameters. The exact confidence intervals are

preferable for average short length of confidence intervals whereas

asymptotic confidence intervals are advisable to use with respect to

coverage probabilities for different choices of the parameters. We proposed

bootstrap confidence intervals and its performance is also quite satisfactory.

173

Whereas to estimate the multi-component stress-strength reliability,

we provided ML and MOM estimators of 1σ and 2σ when both of stress,

strength variates follow the same population. Also, we have estimated

asymptotic confidence interval for multicomponent stress-strength

reliability. The simulation results indicate that in order to estimate the

multicomponent stress-strength reliability for inverse Rayleigh distribution,

the ML method of estimation is preferable than the method of moment

estimation. The length of the confidence interval also decreases as the

sample size increases and coverage probability is close to the nominal value

in all cases for MLE.

174

Table 5.4.1: Average bias of the simulated estimates of R.

1 2( , )σ σ

(n, m) (1,3) (1,2.5) (1,2) (1,1.5) (1,1) (1.5,1) (2,1) (2.5,1) (3,1)

R= 0.10 R= 0.14 R= 0.20 R= 0.31 R= 0.50 R= 0.69 R= 0.80 R= 0.86 R= 0.90

(5,5) 0.0384 0.0414 0.0405 0.0286 -0.0066 -0.0403 -0.0500 -0.0492 -0.0449

0.0149 0.0170 0.0175 0.0129 -0.0036 -0.0193 -0.0224 -0.0207 -0.0179

(10,10) 0.0246 0.0271 0.0269 0.0187 -0.0070 -0.0315 -0.0375 -0.0357 -0.0317

0.0061 0.0069 0.0071 0.0046 -0.0038 -0.0111 -0.0119 -0.0106 -0.0088

(15,15) 0.0223 0.0252 0.0263 0.0212 0.0012 -0.0192 -0.0249 -0.0242 -0.0216

0.0058 0.0069 0.0078 0.0070 0.0016 -0.0043 -0.0057 -0.0054 -0.0046

(20,20) 0.0198 0.0225 0.0235 0.0192 0.0018 -0.0159 -0.0208 -0.0202 -0.0180

0.0035 0.0041 0.0044 0.0034 -0.0010 -0.0050 -0.0056 -0.0050 -0.0042

(25,25) 0.0127 0.0141 0.0139 0.0089 -0.0068 -0.0209 -0.0234 -0.0214 -0.0185

0.0025 0.0029 0.0031 0.0022 -0.0013 -0.0044 -0.0047 -0.0041 -0.0034

In each cell the first row represents the average bias of R using the

MOM and second row represents average bias of R using the MLE.

175

Table 5.4.2: Average MSE of the simulated estimates of R.

1 2( , )σ σ

(n, m) (1,3) (1,2.5) (1,2) (1,1.5) (1,1) (1.5,1) (2,1) (2.5,1) (3,1)

R= 0.10 R= 0.14 R= 0.20 R= 0.31 R= 0.50 R= 0.69 R= 0.80 R= 0.86 R= 0.90

(5,5) 0.0192 0.0255 0.0338 0.0431 0.0490 0.0443 0.0354 0.0274 0.0211

0.0053 0.0082 0.0127 0.0190 0.0236 0.0193 0.0130 0.0084 0.0055

(10,10) 0.0109 0.0154 0.0218 0.0301 0.0362 0.0320 0.0240 0.0172 0.0123

0.0020 0.0033 0.0056 0.0090 0.0118 0.0091 0.0056 0.0033 0.0020

(15,15) 0.0084 0.0121 0.0176 0.0246 0.0294 0.0247 0.0178 0.0123 0.0086

0.0014 0.0023 0.0039 0.0065 0.0085 0.0064 0.0038 0.0022 0.0013

(20,20) 0.0074 0.0107 0.0155 0.0216 0.0256 0.0211 0.0149 0.0101 0.0069

0.0010 0.0016 0.0028 0.0047 0.0063 0.0047 0.0028 0.0016 0.0009

(25,25) 0.0053 0.0079 0.0118 0.0174 0.0218 0.0184 0.0129 0.0086 0.0058

0.0007 0.0012 0.0022 0.0037 0.0050 0.0037 0.0022 0.0012 0.0007

In each cell the first row represents the average MSE of R using the

MOM and second row represents average MSE of R using the MLE.

176

Table 5.4.3: Average confidence length of the simulated 95% confidence intervals of R.

1 2( , )σ σ

(n, m) Interval

(1,3) (1,2.5) (1,2) (1,1.5) (1,1) (1.5,1) (2,1) (2.5,1) (3,1)

R= 0.10 R= 0.14 R= 0.20 R= 0.31 R= 0.50 R= 0.69 R= 0.80 R= 0.86 R= 0.90

(5,5) A 0.2395 0.3049 0.3912 0.4933 0.5612 0.4985 0.3977 0.3111 0.2450

B 0.2228 0.2758 0.3422 0.4161 0.4617 0.4166 0.3430 0.2766 0.2235

C 0.2705 0.3397 0.4272 0.5241 0.5819 0.5200 0.4214 0.3338 0.2650

(10,10) A 0.1627 0.2114 0.2781 0.3607 0.4176 0.3650 0.2832 0.2160 0.1666

B 0.1461 0.1867 0.2403 0.3037 0.3447 0.3040 0.2407 0.1870 0.1464

C 0.1942 0.2467 0.3140 0.3881 0.4211 0.3517 0.2676 0.2021 0.1549

(15,15) A 0.1335 0.1741 0.2301 0.2995 0.3457 0.2981 0.2284 0.1726 0.1322

B 0.1158 0.1496 0.1953 0.2506 0.2871 0.2507 0.1954 0.1497 0.1159

C 0.1532 0.1968 0.2543 0.3201 0.3525 0.2934 0.2211 0.1655 0.1261

(20,20) A 0.1138 0.1490 0.1981 0.2598 0.3021 0.2606 0.1990 0.1499 0.1145

B 0.0988 0.1283 0.1688 0.2184 0.2514 0.2182 0.1685 0.1281 0.0986

C 0.1564 0.1966 0.2463 0.2968 0.3025 0.2428 0.1777 0.1308 0.0986

(25,25) A 0.1012 0.1328 0.1770 0.2330 0.2716 0.2339 0.1781 0.1338 0.1020

B 0.0875 0.1140 0.1506 0.1959 0.2263 0.1957 0.1504 0.1138 0.0873

C 0.1299 0.1645 0.2081 0.2538 0.2659 0.2105 0.1539 0.1131 0.0851

A: Asymptotic confidence interval B: Exact confidence interval

C: Bootstrap confidence interval

177

Table 5.4.4: Average coverage probability of the simulated 95% confidence intervals of R.

1 2( , )σ σ

(n, m) Interval

(1,3) (1,2.5) (1,2) (1,1.5) (1,1) (1.5,1) (2,1) (2.5,1) (3,1)

R= 0.10 R= 0.14 R= 0.20 R= 0.31 R= 0.50 R= 0.69 R= 0.80 R= 0.86 R= 0.90

(5,5) A 0.9337 0.9357 0.9387 0.9423 0.9330 0.9400 0.9397 0.9367 0.9353

B 0.8796 0.8784 0.8765 0.8744 0.8731 0.8765 0.8774 0.8797 0.8803

C 0.9590 0.9590 0.9590 0.9590 0.9590 0.9590 0.9590 0.9590 0.9590

(10,10) A 0.9440 0.9447 0.9463 0.9470 0.9497 0.9533 0.9557 0.9547 0.9540

B 0.8892 0.8882 0.8878 0.8847 0.8819 0.8825 0.8840 0.8864 0.8872

C 0.9500 0.9490 0.9490 0.9490 0.9480 0.9460 0.9460 0.9460 0.9460

(15,15) A 0.9430 0.9407 0.9413 0.9390 0.9387 0.9403 0.9423 0.9440 0.9440

B 0.8886 0.8885 0.8873 0.8863 0.8864 0.8876 0.8883 0.8888 0.8893

C 0.9340 0.9340 0.9340 0.9340 0.9300 0.9300 0.9300 0.9300 0.9310

(20,20) A 0.9440 0.9447 0.9467 0.9463 0.9440 0.9427 0.9443 0.9463 0.9487

B 0.8966 0.8962 0.8954 0.8957 0.8961 0.8941 0.8954 0.8949 0.8963

C 0.8300 0.8310 0.8320 0.8350 0.8360 0.8380 0.8450 0.8430 0.8430

(25,25) A 0.9473 0.9463 0.9473 0.9467 0.9470 0.9473 0.9507 0.9517 0.9523

B 0.8942 0.8938 0.8933 0.8931 0.8931 0.8930 0.8938 0.8948 0.8951

C 0.8080 0.8060 0.8040 0.8000 0.8070 0.8090 0.8130 0.8140 0.8130

A: Asymptotic confidence interval, B: Exact confidence interval and C: Bootstrap confidence interval.

178

Table 5.4.5: Average bias of the simulated estimates of ,s kR .

1 2( , )σ σ

(n, m) (s,k)

(1,3)

(1,3) (1,2.5) (1,2) (1,1.5) (1,1) (1.5,1) (2,1) (2.5,1) (3,1)

,s kR =0.964 ,s kR =0.949

,s kR =0.923 ,s kR =0.871

,s kR =0.750 ,s kR =0.571

,s kR =0.429 ,s kR =0.324

,s kR =0.250

(5,5) -0.0063 -0.0082 -0.0108 -0.0136 -0.0120 0.0011 0.0137 0.0209 0.0236

-0.0239 -0.0296 -0.0367 -0.0437 -0.0402 -0.0127 0.0148 0.0328 0.0423

(10,10) -0.0036 -0.0048 -0.0065 -0.0087 -0.0091 -0.0026 0.0044 0.0087 0.0105

-0.0128 -0.0165 -0.0215 -0.0271 -0.0262 -0.0075 0.0119 0.0244 0.0306

(15,15) -0.0022 -0.0029 -0.0039 -0.0053 -0.0055 -0.0012 0.0033 0.0060 0.0070

-0.0106 -0.0133 -0.0166 -0.0198 -0.0166 0.0006 0.0166 0.0260 0.0301

(20,20) -0.0012 -0.0016 -0.0022 -0.0028 -0.0022 0.0016 0.0051 0.0069 0.0074

-0.0087 -0.0113 -0.0149 -0.0191 -0.0191 -0.0060 0.0079 0.0169 0.0212

(25,25) -0.0015 -0.0020 -0.0028 -0.0040 -0.0048 -0.0028 0.0002 0.0016 0.0025

-0.0073 -0.0097 -0.0128 -0.0166 -0.0164 -0.0044 0.0081 0.0158 0.0193

(n, m)

(2,4) ,s kR =0.938 ,s kR =0.913

,s kR =0.869 ,s kR =0.784

,s kR =0.600 ,s kR =0.366

,s kR =0.214 ,s kR =0.127

,s kR =0.077

(5,5) -0.0102 -0.0129 -0.0159 -0.0173 -0.0067 0.0186 0.0321 0.0335 0.0294

-0.0362 -0.0432 -0.0504 -0.0525 -0.0289 0.0241 0.0560 0.0658 0.0636

(10,10) -0.0059 -0.0077 -0.0099 -0.0118 -0.0076 0.0066 0.0150 0.0163 0.0143

-0.0204 -0.0255 -0.0313 -0.0345 -0.0198 0.0182 0.0411 0.0471 0.0443

(15,15) -0.0036 -0.0046 -0.0060 -0.0072 -0.0044 0.0048 0.0101 0.0107 0.0092

-0.0163 -0.0197 -0.0232 -0.0237 -0.0087 0.0231 0.0401 0.0428 0.0387

(20,20) -0.0020 -0.0026 -0.0032 -0.0035 -0.0006 0.0067 0.0102 0.0100 0.0083

-0.0140 -0.0176 -0.0220 -0.0249 -0.0152 0.0122 0.0289 0.0330 0.0308

(25,25) -0.0025 -0.0033 -0.0044 -0.0057 -0.0050 0.0003 0.0039 0.0048 0.0043

-0.0119 -0.0152 -0.0191 -0.0217 -0.0127 0.0122 0.0267 0.0297 0.0270

In each cell the first row represents the average bias of ,s kR using the MLE

and second row represents average bias of ,s kR using the MOM.

179

Table 5.4.6: Average MSE of the simulated estimates of ,s kR .

1 2( , )σ σ

(n, m) (s,k)

(1,3)

(1,3) (1,2.5) (1,2) (1,1.5) (1,1) (1.5,1) (2,1) (2.5,1) (3,1)

,s kR =0.964 ,s kR =0.949

,s kR =0.923 ,s kR =0.871

,s kR =0.750 ,s kR =0.571

,s kR =0.429 ,s kR =0.324

,s kR =0.250

(5,5) 0.0010 0.0017 0.0033 0.0069 0.0148 0.0216 0.0219 0.0190 0.0153

0.0072 0.0105 0.0158 0.0248 0.0389 0.0478 0.0482 0.0449 0.0400

(10,10) 0.0004 0.0007 0.0014 0.0033 0.0079 0.0122 0.0122 0.0102 0.0079

0.0024 0.0040 0.0071 0.0131 0.0246 0.0334 0.0341 0.0310 0.0267

(15,15) 0.0002 0.0004 0.0008 0.0020 0.0050 0.0079 0.0078 0.0064 0.0048

0.0028 0.0042 0.0068 0.0118 0.0208 0.0277 0.0280 0.0251 0.0212

(20,20) 0.0001 0.0003 0.0006 0.0014 0.0037 0.0060 0.0061 0.0050 0.0038

0.0014 0.0024 0.0044 0.0086 0.0171 0.0240 0.0244 0.0219 0.0185

(25,25) 0.0001 0.0002 0.0004 0.0011 0.0028 0.0046 0.0046 0.0037 0.0028

0.0011 0.0019 0.0036 0.0074 0.0152 0.0216 0.0217 0.0190 0.0156

(n, m) (2,4) ,s kR =0.938 ,s kR =0.913

,s kR =0.869 ,s kR =0.784

,s kR =0.600 ,s kR =0.366

,s kR =0.214 ,s kR =0.127

,s kR =0.077

(5,5) 0.0026 0.0045 0.0082 0.0153 0.0265 0.0286 0.0218 0.0142 0.0087

0.0152 0.0212 0.0304 0.0436 0.0580 0.0594 0.0521 0.0422 0.0328

(10,10) 0.0010 0.0019 0.0037 0.0078 0.0151 0.0163 0.0114 0.0067 0.0037

0.0060 0.0096 0.0158 0.0266 0.0409 0.0434 0.0360 0.0271 0.0194

(15,15) 0.0006 0.0011 0.0022 0.0048 0.0097 0.0106 0.0071 0.0039 0.0020

0.0063 0.0092 0.0142 0.0225 0.0338 0.0356 0.0289 0.0210 0.0146

(20,20) 0.0004 0.0008 0.0016 0.0035 0.0073 0.0083 0.0056 0.0031 0.0016

0.0037 0.0060 0.0103 0.0181 0.0294 0.0314 0.0252 0.0183 0.0128

(25,25) 0.0003 0.0006 0.0012 0.0027 0.0056 0.0062 0.0041 0.0022 0.0011

0.0028 0.0049 0.0087 0.0160 0.0267 0.0280 0.0216 0.0149 0.0098

In each cell the first row represents the average MSE of ,s kR using the MLE

and second row represents average MSE of ,s kR using the MOM.

180

Table 5.4.7: Average confidence length and coverage probability of the simulated 95% confidence intervals of ,s kR using MLE.

(n, m) (s,k)

(1,3)

1 2( , )σ σ

(1,3) (1,2.5) (1,2) (1,1.5) (1,1) (1.5,1) (2,1) (2.5,1) (3,1)

(5,5) A 0.09907 0.13539 0.19354 0.29058 0.44710 0.55454 0.55611 0.51087 0.45178

B 0.9413 0.9413 0.9413 0.9423 0.9470 0.9483 0.9413 0.9377 0.9413

(10,10) A 0.06476 0.08956 0.13033 0.20109 0.32180 0.40908 0.41050 0.37338 0.32572

B 0.9410 0.9417 0.9430 0.9440 0.9457 0.9440 0.9493 0.9493 0.9483

(15,15) A 0.05248 0.07278 0.10637 0.16525 0.26696 0.34022 0.33943 0.30609 0.26471

B 0.9503 0.9503 0.9503 0.9500 0.9483 0.9480 0.9450 0.9447 0.9470

(20,20) A 0.04448 0.06181 0.09065 0.14166 0.23108 0.29669 0.29646 0.26709 0.23056

B 0.9510 0.9510 0.9500 0.9500 0.9463 0.9457 0.9510 0.9500 0.9500

(25,25) A 0.03934 0.05472 0.08038 0.12597 0.20659 0.26653 0.26668 0.24011 0.20699

B 0.9490 0.9497 0.9500 0.9493 0.9490 0.9507 0.9520 0.9537 0.9547

(n, m) (2,4) 1 2( , )σ σ

(1,3) (1,2.5) (1,2) (1,1.5) (1,1) (1.5,1) (2,1) (2.5,1) (3,1)

(5,5) A 0.16630 0.22371 0.31136 0.44463 0.61153 0.63405 0.53166 0.41080 0.30794

B 0.9413 0.9413 0.9413 0.9440 0.9463 0.9457 0.9400 0.9307 0.9287

(10,10) A 0.10985 0.14998 0.21336 0.31467 0.45127 0.47469 0.39039 0.29141 0.20993

B 0.9417 0.9423 0.9437 0.9460 0.9433 0.9510 0.9510 0.9477 0.9443

(15,15) A 0.08923 0.12225 0.17486 0.26002 0.37643 0.39440 0.31893 0.23308 0.16441

B 0.9503 0.9503 0.9487 0.9500 0.9483 0.9467 0.9487 0.9473 0.9447

(20,20) A 0.07576 0.10408 0.14953 0.22396 0.32790 0.34567 0.27889 0.20265 0.14204

B 0.9510 0.9507 0.9500 0.9483 0.9473 0.9520 0.9517 0.9473 0.9450

(25,25) A 0.06706 0.09224 0.13279 0.19963 0.29421 0.31163 0.25118 0.18176 0.12672

B 0.9497 0.9497 0.9503 0.9483 0.9520 0.9510 0.9563 0.9570 0.9547

A: Average confidence length B: Coverage probability