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Journal of Reliability and Statistical Studies; ISSN (Print): 0974-8024, (Online):2229-5666

Vol. 8, Issue 2 (2015): 41-58

BAYESIAN ESTIMATION OF RELIABILITY IN TWO-

PARAMETER GEOMETRIC DISTRIBUTION

Sudhansu S. Maiti

*

1 and Sudhir Murmu 2

1Department of Statistics, Visva-Bharati University, Santiniketan, India

2District Rural Development Agency, Khunti, Jharkand, India

E Mail:

*

1dssm1@rediffmail.com, 2sudhir.murmu@yahoo.com

Received February 22, 2015

Modified July 07, 2015

Accepted July 27, 2015

Abstract

Bayesian estimation of reliability of a component,

)()( tXPtR

≥

=

, when

X

follows two-parameter geometric distribution, has been considered. Maximum Likelihood

Estimator (MLE), an Unbiased Estimator and Bayesian Estimator have been compared. Bayesian

estimation of component reliability )( YXPR

≤

=

, arising under stress-strength setup, when

Y

is assumed to follow independent two-parameter geometric distribution has also been

discussed assuming independent priors for parameters under different loss functions.

Key Words

: ML Estimator, Quasi-Bayes Estimate, Unbiased Estimator.

1. Introduction

Various lifetime models have been proposed to describe the important

characteristics of lifetime data. Most of these models assume lifetime to be a

continuous random variable. However, it is sometimes impossible or inconvenient to

measure the life length of a device on a continuous scale. In practice, we come across

situations where lifetimes are recorded on a discrete scale. Discrete life distributions

have been suggested and properties have been studied by Barlow and Proschan [1].

Here one may consider lifetime to be the number of successful cycles or operations of a

device before failure. For example, the bulb in Xerox machine lights up each time a

copy is taken. A spring may breakdown after completing a certain number of cycles of

‘to-and-fro’ movements.

The study of discrete distributions in lifetime models is not very old. Yakub

and Khan [11] considered the geometric distribution as a failure law in life testing and

obtained various parametric and nonparametric estimation procedures for reliability

characteristics. Bhattacharya and Kumar [2] have considered the parametric as well as

Bayesian approach to the estimation of the mean life cycle and that of reliability

function for complete as well censored sample. Krishna and Jain [3] obtained classical

and Bayes estimation of reliability for some basic system configurations.

The geometric distribution [abbreviated as

)(

θ

Geo

] is given by

=−== xxXP

x

;)1()(

θθ

0, 1, 2,……..;

10

<

<

θ

42 Journal of Reliability and Statistical Studies, December 2015, Vol. 8(2)

and the component reliability is given by

( ) ;

t

R t t

θ

= =

0, 1, 2,……….. (1)

Modeling in terms of two-parameter geometric and estimation of its

parameters and related functions are of special interest to a manufacturer who wishes to

offer a minimum warranty life cycle of the items produced.

The two-parameter geometric distribution abbreviated as

),(

θ

rGeo

] is given by

;)1()(

rx

xXP

−

−==

θθ

,....2,1,

+

+

=

rrrx

10

<

<

θ

and

∈

r

{0, 1, 2,…..}

and the component reliability is given by

;)(

rt

tR

−

=

θ

,.........2,1,

+

+

=

rrrt

(2)

The continuous counterpart of geometric (i.e. exponential) distribution, one-parameter

as well as two-parameter is considered by a host of authors (see ref. Sinha [10]).

In the stress-strength setup,

)( YXPR

≤

=

originated in the context of the

reliability of a component of strength

Y

subjected to a stress

X

. The component fails if

at any time the applied stress is greater than its strength and there is no failure when

Y

X

≤

. Thus

R

is a measure of the reliability of the component. Many authors

considered the problem of estimation of

R

in continuous setup in the past. In the

discrete setup, a limited work has been done so far. Maiti [4] has considered stress (or

demand)

X

and strength (or supply)

Y

as independently distributed geometric random

variables, whereas Sathe and Dixit [9] assumed as negative binomial variables, and

derived both MLE and UMVUE of

.R

Maiti and Kanji [7] has derived some

expressions of

R

using a characterization of

)( YXP

≤

and Maiti ([5], [6])

considered MLE, UMVUE and Bayes Estimation of R for some discrete distributions

useful in life testing.

If

X

and

Y

follow two-parameter geometric distributions with parameters

),(

11

r

θ

and

),(

22

r

θ

respectively, then

δ

ρθ

2

=R

for

0

>

δ

δ

θρ

−

−−=

1

)1(1

for

,0

<

δ

(3)

where

21

1

1

1

θθ

θ

ρ

−

−

=

and

.

21

rr −=

δ

The objective of this article is to compare the estimates of reliability for mission

time as well as stress-strength set up for two-parameter geometric distribution. The

paper is organized as follows. In section 2, we have summarized MLE, Unbiased

Estimator (c.f. Maiti et al. [8]) and derived Quasi-Bayes estimate of

)(tR

under

different loss functions assuming conjugate priors for the parameters involved. We

attempt Quasi-Bayes estimate since derivation of posterior distribution seems to be

intractable. Different scale invariant loss functions are considered, viz., squared error

loss, squared log error loss, Modified Linear Exponential (MLINEX) loss, Absolute

Bayesian estimation of reliability in two-parameter … 43

error loss and the corresponding estimates have been compared. We have discussed

MLE, Unbiased Estimator and Quasi-Bayes estimates of

R

in section 3. All

comparisons have been made through simulation study in section 4. Section 5

concludes.

2. Inference on

)(tR

Let

),....,,(

21

n

XXX

be a random sample from

),(

θ

rGeo

. Maximum Likelihood

Estimator of

r

and

θ

are

)1(

X

and

S

n

S

+

respectively, where

)1(

X

is smallest

observation among

n

XXX ,....,,

21

and

.)(

1)1(

∑

=

−=

n

ii

XXS

ML Estimators of

)(tR

is given by

1)(

ˆ=tR

M

for

)1(

Xt

≤

)1(

Xt

Sn

S

−

+

=

for

.

)1(

Xt

>

Here

(

)

SX ,

)1(

is sufficient statistic for

),(

θ

r

, but it is not complete (c.f. Maiti et al.

[8]).

For

,1

=

n

(

)

1,|

)1(

=

SXxf

For

2

=

n

(

)

2

1

,|

)1(

=SXxf

if

)1(

Xx

=

2

1

=

if

SXx

+

=

)1(

For

,3

≥

n

,nS

<

( )

(

)

( )

−+

−−

−+−−

=S

nS

XxS

nXxS

SXxf 1

2

,|

)1(

)1(

)1(

if

SXxX

+

≤

≤

)1()1(

= 0 otherwise

For

,,3 nSn

≥

≥

( )

−

−

−

−+

−+

=1

112

,|

)1(

n

S

S

nS

S

nS

SXxf

if

)1(

Xx

=

.

44 Journal of Reliability and Statistical Studies, December 2015, Vol. 8(2)

(

)

( )

(

)

−

−

−

−+

−

−−−

−

−−

−+−−

=1

11

2

1

2

)1(

)1(

)1(

n

S

S

nS

n

XxS

XxS

nXxS

if

(

)

1

)1()1(

−

−

+

≤

<

nSXxX

(

)

( )

−

−

−

−+

−−

−+−−

=1

11

2

)1(

)1(

n

S

S

nS

XxS

nXxS

if

(

)

SXxnSX

+

≤

<

−

−

+

)1()1(

1

= 0 Otherwise.

Hence, using the Rao-Blackwell theorem, an unbiased estimator of

)(tR

is given as

follows:

For

,1

=

n

(

)

1

ˆ=tR

U

if

)1(

Xt

≤

0

=

if

.

)1(

Xt

>

For

,2

=

n

(

)

1

ˆ=tR

U

if

)1(

Xt

≤

2

1

= if

SXtX

+

≤

<

)1()1(

0

=

if

SXt

+

>

.

)1(

.

For

3

≥

n

and

,nS

<

(

)

1

ˆ=tR

U

if

)1(

Xt

≤

(

)

( )

∏

∑

−

=

+

=

−−++

−−++

−++

−

=

2

1)1(

)1(

)1(

1

1

1

1

)1(

n

j

SX

tx

jnSX

jxnSX

nSX

n

if

SXtX

+

≤

<

)1()1(

= 0 if

..

)1(

SXt

+

>

For

3

≥

n

and

,nS

≥

(

)

1

ˆ=tR

U

if

)1(

Xt

≤

(

)

( )

(

)

(

)

∑

−−+

=

−

−

−

−+

−

−−−

−

−−

−+−−

=

1

)1(

)1(

)1(

)1(

1

11

2

1

2

nSX

tx

n

S

S

nS

n

XxS

XxS

nXxS

Bayesian estimation of reliability in two-parameter … 45

(

)

( )

( )

∑

+

+−−=

−

−

−

−+

−−

−+−−

+

SX

nXx

n

S

S

nS

XxS

nXxS

)1(

)1(

11 )1(

)1(

1

11

2

if

(

)

1

)1()1(

−

−

+

≤

<

nSXtX

=

(

)

( )

∑

+

=

−

−

−

−+

−−

−+−−

SX

tx

n

S

S

nS

XxS

nXxS

)1(

1

11

2

)1(

)1(

if

(

)

SXtnSX

+

≤

<

−

−

+

)1()1(

1

= 0 Otherwise.

We obtain the Bayes estimates of the parameters under the assumptions that

the parameters

θ

and

r

are random variables. It is assumed that

θ

and

r

have

independent beta and Poisson priors as follows:

( ) ( ) ( )

;1

,

1

1

1

−

−

−=

q

p

qpB

θθθπ

,10

<

<

θ

0,

>

qp

and

( )

;

!

r

er

r

λ

φ

λ

−

=

=

r

0, 1, 2…

Here

( ) ( )

∫

−

−

−=

1

0

1

1

.1,

θθθ

dqpB

q

p

The joint distribution of

θ

and

r

given observations is given by

(

)

(

)

(

)

θπθθ

,.,||, rrxLXrg =

( )

( )

( ) ( )

!

.1

,

1

.1

1

1

1

r

e

qpB

r

q

p

rx

n

n

ii

λ

θθθθ

λ

−

−

−

−

−

∑

−=

=

( )

( )

( )

;

!

1|,

1

1

)1(

r

eXrg

r

qn

prXnS

λ

θθθ

λ

−

−+

−+−+

−∝

,10

<

<

θ

)1(

,.......2,1,0 Xr

=

.

The posterior distributions of

θ

and

r

are as follows:

( ) ( )( )

( )( )

∑

=

−

−

++−+

++−+

=

)1(

0)1(

)1(

)1(1

,

!

,

!

,|

X

w

w

r

qnpwXnSB

w

e

qnprXnSB

r

e

SXrg

λ

λ

λ

λ

and

46 Journal of Reliability and Statistical Studies, December 2015, Vol. 8(2)

(

)

(

)

×−=

−+

−++ 1

1

)1(2

1,|

)1(

qn

pnXS

SXg

θθθ

(

)

( )( )

∑

∑

=

−

=

−

++−+

=

)1(

)1(

0)1(

0

,

!

!

X

w

w

X

w

w

n

qnpwXnSB

w

e

w

e

λ

θλ

λ

λ

Derivation of posterior distribution of

)(tR

seems to be intractable.

Therefore, Quasi-Bayes estimates of these expressions by substituting Bayes estimates

of the parameters involved have been obtained. Comparisons have been made through

simulation study. Some scale invariant loss functions have been considered to get

Bayes estimates of

θ

and

r

. These are summarized in the following discussion.

2.1 Squared Error Loss

The loss function is given by

( )

2

1

1,

−=

α

δ

δα

L

and estimates of

θ

and

r

are

( )( )

( )( )

∑

∑

=

−

=

−

+−+−+

+−+−+

=

=

)1(

)1(

0)1(

0)1(

)1(

2

,2

!

,1

!

,|.

1

ˆ

X

w

w

X

w

w

qnpwXnSB

w

e

qnpwXnSB

w

e

SXE

λ

λ

θ

θ

θ

λ

λ

and

( )( )

( )( )

∑

∑

=

−

=

−

++−+

++−+

=

=

)1(

)1(

1)1(

2

1)1(

)1(

2

,

!

1

,

!

1

,|

1

ˆ

X

w

w

X

w

w

qnpwXnSB

w

e

w

qnpwXnSB

w

e

w

SXr

r

Er

λ

λ

λ

λ

respectively.

2.2. Squared log error loss

The loss function is given by

( ) ( )

2

2

2

lnlnln,

=−=

α

δ

αδδα

L

and estimates of

θ

and

r

are

Bayesian estimation of reliability in two-parameter … 47

(

)

,

ˆ

,|ln

)1(

SXE

e

θ

θ

=

where

( )

( )

( )( )

∑

∑∫

=

−

=

−+

−++

−

++−+

−

=

)1(

)1( )1(

0)1(

0

1

0

1

1

)1(

,

!

1.ln

!

,|ln

X

w

w

X

w

qn

pnXS

w

qnpwXnSB

w

e

d

w

e

SXE

λ

θθθθ

λ

θ

λ

λ

( )( )

( )

( )( )

∑

∑

=

−

=

−

++−+

×++−+

=

)1(

)1(

0)1(

0)1(

,

!

,.

!

X

w

w

X

w

w

qnpwXnSB

w

e

wIqnpwXnSB

w

e

λ

λ

λ

λ

where

(

)

(

)

(

)

(

)

(

)

[

]

nqpwXnSpwXnSwI

+

+

+

−

+

−

+

−

+

=

)1()1(

ψ

ψ

and

(

)

u

ψ

is digamma function.

and

(

)

,

ˆ

,|ln

)1(

SXrE

er =

where

( ) ( )( )

( )( )

∑

∑

=

−

=

−

++−+

++−+

=

)1(

)1(

0)1(

1)1(

)1(

,

!

,.

!

.ln

,|ln

X

w

w

X

w

w

qnpwXnSB

w

e

qnpwXnSB

w

ew

SXrE

λ

λ

λ

λ

2.3 MLINEX

The loss function is given by

( )

0,0;1ln,

3

>≠

−−

=ccL

γ

α

δ

γ

α

δ

δα

γ

and estimates of

θ

and

r

[assuming

1

=

c

] are

( )

[

]

,,|

ˆ

1

)1(

γ

γ

θθ

−

−

=SXE

where

( )

( )( )

( )( )

∑

∑

=

−

=

−

−

++−+

+−+−+

=

)1(

)1(

0)1(

0)1(

)1(

,

!

,

!

,|

X

w

w

X

w

w

qnpwXnSB

w

e

qnpwXnSB

w

e

SXE

λ

γ

λ

θ

λ

λ

γ

48 Journal of Reliability and Statistical Studies, December 2015, Vol. 8(2)

and

( )

[

]

,,|

ˆ

1

)1(

γ

γ

−

−

=SXrEr

where

( )

( )( )

( )( )

∑

∑

=

−

=

−−

−

++−+

++−+

=

)1(

)1(

0)1(

0)1(

)1(

,

!

,

!

.

,|

X

w

w

X

w

w

qnpwXnSB

w

e

qnpwXnSB

w

ew

SXrE

λ

λ

λ

λγ

γ

Particular Case: when

,1

=

γ

we have Entropy loss function. Then

( )( )

( )( )

∑

∑

=

−

=

−

+−+−+

++−+

=

)1(

)1(

0)1(

0)1(

,1

!

,

!

ˆ

X

w

w

X

w

w

qnpwXnSB

w

e

qnpwXnSB

w

e

λ

λ

θ

λ

λ

and

( )( )

( )( )

∑

∑

=

−

=

−

+−+−+

++−+

=

)1(

)1(

0)1(

0)1(

,2

!

,

!

ˆ

X

w

w

X

w

w

qnpwXnSB

w

e

qnpwXnSB

w

e

r

λ

λ

λ

λ

2.4 Absolute error loss

The loss function is given by

( )

α

δ

δα

−= 1,

4

L

Then M=

θ

ˆ

such that

( )

( )

2

1

,|

1

,|

1

1

0

)1(2

0

)1(2

=

∫

∫

θθ

θ

θθ

θ

dSXg

dSXg

M

i.e.

( )

( )

( )( )

2

1

,1

!

1

!

)1(

)1(

)1(

0)1(

1

0

2

0

=

+−+−+

−

∑

∫

∑

=

−

−+

−+−+

=

−

X

w

w

qn

M

prXnS

w

X

w

qnpwXnSB

w

e

d

w

e

λ

θθθ

λ

λ

λ

To get the Bayes estimate of

r

, we have to solve the following equation for

.M

( )( )

( )( )

.

2

1

,

!

.

1

,

!

.

1

)1(

)

1)1(

)1(

1

=

++−+

++−+

∑

∑

=

−

=

−

X

w

w

w

M

w

qnpwXnSB

w

e

w

qnpwXnSB

w

e

w

λ

λ

λ

λ

Bayesian estimation of reliability in two-parameter … 49

3. Inference on R

Let

(

)

1

,......,

21 n

XXX

and

(

)

2

,......,,

21 n

YYY

be random samples from

(

)

11

,

θ

rGeo and

(

)

22

,

θ

rGeo respectively.

(

)

1)1(

,SX

and

(

)

2)1(

,SY

are defined in

the same way as in section 2. Hence ML Estimator of

R

is given by

δ

ρ

ˆ

22

2

ˆ

ˆ

+

=Sn

S

R

M

for

0

ˆ

>

δ

( )

δ

ρ

ˆ

11

1

ˆ

11

−

+

−−= Sn

S

for

,0

ˆ<

δ

where

122121

2121

ˆSnSnnn

Snnn

++

+

=

ρ

and

.

ˆ

)1()1(

YX −=

δ

Application of the Rao-Blackwell theorem gives an unbiased estimator of

R

as

( )

(

)

( ) ( )

2)1(1)1(

,min

1

1)1(

1

,|.,|,|

1

ˆ

221

)1(

)1(

)1(

SYyfSXxfSXxf

n

R

W

xy

WW

Yx

Y

Xx

U

∑∑∑

==+=

++=

if

)1()1(

YX

<

(

)

( ) ( )

2)1(1)1(

,min

1

,|.,|

1

221

)1(

SYyfSXxf

n

W

xy

WW

Yx

∑∑

==

+=

if

)1()1(

YX

=

( )

(

)

( ) ( )

2)1(1)1(

,min

1

2)1(

1

,|.,|,|

1

221

)1(

2

)1(

SYyfSXxfSYyf

n

W

xy

WW

Xx

W

Xy

∑∑∑

=+==

++=

if

)1()1(

YX

>

where,

1)1(1

SXW

+

=

,

.

2)1(2

SYW

+

=

The variance of this unbiased estimator will

be smaller than the unbiased estimator

(

)

.

1

1 2

1 1

21

∑ ∑

= =

<

n

i

n

jji

YXI

nn

Derivation of

posterior distribution of

R

in this case seems to be intractable. Therefore, Quasi-Bayes

estimates of

R

by substituting Bayes estimates of the parameters derived in section 2

have been found out.

4. Simulation and Discussion

We generate sample of size

n

and on the basis of this sample, calculate MLE

and UE and Quasi-Bayes estimate of

(

)

tR and their Mean Squared Errors (MSEs).

10000 such estimates have been calculated and results, on the basis of these estimates

have been reported in Tables 1-4. In each table, there are six rows in average estimate

of reliability and their MSEs. Information reported as 1st row: MLE, 2

nd

row: UE, 3

rd

row: Bayes estimate under squared error loss, 4

th

row: Bayes estimate under squared log

50 Journal of Reliability and Statistical Studies, December 2015, Vol. 8(2)

error, 5

th

row: Bayes estimate under entropy loss, 6

th

row: Bayes estimate under

Absolute error loss. Each table has been prepared considering different choices of a

particular parameter, keeping others fixed at initial set up. All simulations and

calculations have been done using R-Software and algorithms used can be obtained by

contacting the corresponding author.

Maiti et al. [8] have reported that MLE of

(

)

tR perform better in mean square

error sense if

5.0)(02.0

<

<

tR

; otherwise UE performs well. From tables 1-4, it is

observed that quasi-Bayes estimates are better than MLE as well as UE in all most all

cases. Among four quasi-Bayes estimates, the estimate under squared error loss seems

better, whereas the estimate under absolute error, the performance is not encouraging.

We also generate samples of size

1

n and

2

n, and on the basis of these

samples, calculate MLE and UE and Quasi-Bayes estimate of

R

and their Mean

Squared Errors (MSEs). Here, we take ,10

21

== nn and 1000 estimates of

R

have

been taken for calculating MSEs [Tables 5-6]. Maiti et al. [8] have reported that UE of

R

perform better in mean square error sense for extreme low and high reliable

components; in other cases, MLE is better. But in all most all case, quasi-Bayes

estimates are better.

5.

Concluding Remark

This paper takes into account the Bayes estimation aspect of reliability with

two-parameter geometric lifetime. The continuous distributions are widely referenced

probability laws used in reliability and life testing for continuous data. When the lives

of some equipment and components are being measured by the number of completed

cycles of operations or strokes, or in case of periodic monitoring of continuous data, the

discrete distribution is a natural choice. Bayesian Estimation procedures have been

worked out for estimating reliability assuming independent priors for parameters under

different loss functions. In all most all cases, quasi-Bayes estimates of reliability are

better in mean square error sense. Some other distributions that are used as discrete life

distributions are to be considered and their Bayes estimation aspect of reliability are to

be attempted in future.

Acknowledgement

The authors would like to thank the referees for very careful reading of the

manuscript and making a number of nice suggestions which improved the earlier

version of the manuscript.

References

1. Barlow, R.E. and Proschan, F. (1967). Mathematical Theory of Reliability, John Willy &

Sons, Inc., New York.

2. Bhattacharya, S.K. and Kumar, S. (1985). Discrete life testing, IAPQR Transactions, 13,

p. 71-76.

3. Krishna, H. and Jain, N. (2002). Classical and Bayes Estimation of Reliability

Characteristics of some Basic System Configurations with Geometric Lifetimes of

Components, IAPQR Transactions, 27, p. 35-49.

Bayesian estimation of reliability in two-parameter … 51

4. Maiti, S.S. (1995). Estimation of

(

)

YXP ≤

in the geometric case, Journal of the

Indian Statistical Association, 33, p. 87-91.

5. Maiti, S.S. (2005). Bayesian Estimation of

(

)

YXP ≤

for some Discrete Life

Distributions, Contributions to Applied and Mathematical Statistical, 3, p. 81-96.

6. Maiti, S.S. (2006). On Estimation of

(

)

YXP ≤

for Discrete Distributions useful in

Life Testing, IAPQR Transactions, 31, p. 39-46.

7. Maiti, S.S. and Kanji, A. (2005). A note on Characterization of

(

)

YXP ≤

for Discrete

Life Distributions, Journal of Applied Statistical Science, 14, p. 275-279.

8. Maiti, S.S., Murmu, S. and Chattopadhyay, G. (2015). Inference on Reliability for Two-

parameter Geometric Distribution,

International Journal of Agricultural and

Statistical Sciences, 11(2)

(to appear).

9. Sathe, Y.S. and Dixit, U.J. (2001). Estimation of

(

)

YXP ≤

in the negative binomial

distribution, Journal of Statistical Planning and Inference, 93, p. 83-92.

10. Sinha, S.K. (1986). Reliability and Life Testing, Wiley Eastern Limited, New Delhi.

11. Yakub, M. and Khan, A.H. (1981). Geometric failure law in life testing, Pure and

Applied Mathematika Science, 14, p. 69-76.

t Reliability Avg. Estimates MSE

0.3166573 0.006389343

0.324255 0.00666498

20 0.32768 0.3196006 0.005297303

0.3256051 0.005139682

0.3236131 0.5139682

0.3328666 0.005644666

0.1090191 0.002340292

0.1079153 0.002589489

25 0.1073742 0.1035126 0.002289946

0.1071390 0.002308059

0.1059272 0.002299524

0.1111683 0.002561238

0.03964109 0.0007162865

0.03618544 0.0007241272

30 0.03518437 0.03638538 0.0005862288

0.03813851 0.0006180215

0.03754956 0.0006067526

0.04018129 0.0007150897

0.01482919 0.000193777

0.01198529 0.0001630661

35 0.0115292 0.01269303 0.0001348279

0.01346045 0.0001460612

0.01320140 0.0001421481

0.01440264 0.0001740828

Table 1: Average Estimates and Mean Square Errors of

)(tR

with n=20, r=15, p=8,

q=2.

52 Journal of Reliability and Statistical Studies, December 2015, Vol. 8(2)

n Reliability Avg. Estimates MSE

0.1060938 0.00479161

0.1265446 0.007624039

10 0.1073742 0.09549643 0.003795114

0.1042998 0.003998701

0.1013891 0.003922511

0.1330134 0.009238643

0.1084599 0.003447145

0.1217706 0.004772602

15 0.1073742 0.1045881 0.003014518

0.1098457 0.003102323

0.1080907 0.003068467

0.1185306 0.004424606

0.1086262 0.002678777

0.1184946 0.003431893

20 0.1073742 0.1041115 0.002489043

0.1079099 0.002514939

0.1066405 0.002503606

0.1121600 0.002841681

0.1080681 0.002088382

0.1158652 0.002549381

25 0.1073742 0.1050123 0.001967833

0.1080312 0.001982178

0.1070225 0.001975586

0.1114314 0.002153514

Table 2:

Average Estimates and Mean Square Errors of

)(tR

with r=15, t=25, p=8, q=2.

Bayesian estimation of reliability in two-parameter … 53

r Reliability Avg. Estimates MSE

0.0146542 0.000238691

0.01405295 0.0002928919

5 0.01152922 0.01356151 0.0001464932

0.01438816 0.0001601152

0.01410971 0.00015539861

0.01536586 0.0001893861

0.0384844 0.000921849

0.04098456 0.001267719

10 0.03518437 0.03600980 0.0006123444

0.03781796 0.0006448308

0.03721048 0.0006333065

0.03980784 0.0007365458

0.1086646 0.003386037

0.1220295 0.004699975

15 0.1073742 0.105009 0.002266010

0.1088256 0.002296038

0.107502 0.002283241

0.1130798 0.002605971

0.3177951 0.008842791

0.35122351 0.01040297

20 0.32768 0.3101675 0.006174129

0.3165970 0.005882643

0.3144629 0.005970977

0.3250680 0.006599251

Table 3: Average Estimates and Mean Square Errors of

)(tR

with n=20, t=25, p=8, q=2.

54 Journal of Reliability and Statistical Studies, December 2015, Vol. 8(2)

θ

Reliability Avg. Estimates MSE

0.03177319 0.000551296

0.02824752 0.03256912 0.0006866504

0.7 0.02770768 0.000340732

0.02991801 0.0003669113

p=7, q=3 0.02917032 0.0003570654

0.03059155 0.000396458

0.1069406 0.002548496

0.1073742 0.1166341 0.003241387

0.8 0.1060890 0.002650470

0.1098881 0.02693512

p=8, q=2 0.1086186 0.002676484

0.1142395 0.003059418

0.3413777 0.006712354

0.3486784 0.3679966 0.007528894

0.9 0.3315982 0.006204844

0.3363357 0.006150177

p=9, q=1 0.3347966 0.006167038

0.3841432 0.01139207

0.4763589 0.006831021

0.4839823 0.5050617 0.007228364

0.93 0.4655819 0.005150811

0.4723362 0.005180253

P=13, q=1 0.4702358 0.005167441

0.5644204 0.1685316

0.96

0.6648326

p=24, q=1

0.6677073 0.005206639

0.6918656 0.005490035

0.6256307 0.004290399

0.6394016 0.003793488

0.6353227 0.00390119

0.953012 0.1613833

0.99

0.9043382

p=99, q=1

0.9352147 0.00497277

0.9412775 0.002639834

0.8405322 0.004131144

0.8572153 0.002276667

0.8522604 0.002768958

0.980264 0.230441

Table 4: Average Estimates and Mean Square Errors of

)(tR

with n=20, r=15, t=25.

Bayesian estimation of reliability in two-parameter … 55

12

|rr

5 10

Reliability Avg.

Estimates

MSE Reliability Avg.

Estimates

MSE

0.593202 0.013734 0.103300 0.004158

0.589705 0.013978 0.100187 0.004813

5 0.588235 0.593160 0.008214 0.098864 0.088356 0.002387

0.589218 0.007796 0.094898 0.002323

0.590506 0.007942 0.092725 0.002338

0.589435 0.008925 0.094697 0.002533

0.928161 0.002695 0.593680 0.012493

0.933117 0.003082 0.590026 0.012775

10 0.930794 0.935221 0.001711 0.588235 0.584252 0.008245

0.929792 0.001753 0.580510 0.007879

0.931604 0.001735 0.581726 0.008002

0.929551 0.001922 0.580201 0.009766

0.984090 0.000344 0.925457 0.002877

0.988884 0.000283 0.930138 0.003267

15 0.988368 0.987664 0.000160 0.930794 0.932667 0.001880

0.985964 0.000186 0.927189 0.001946

0.986543 0.000176 0.929023 0.001919

0.985866 0.000205 0.927168 0.002199

0.995580 6.22

5

10

−

×

0.983920 0.000375

0.997830 3.44

5

10

−

×

0.988676 0.000319

20 0.998045 0.997158 2.39

5

10

−

×

0.988368 0.987053 0.000177

0.996679 2.90

5

10

−

×

0.985314 0.000207

0.996845 2.72

5

10

−

×

0.985909 0.000196

0.996672 3.18

5

10

−

×

0.985451 0.000232

Table 5a: Average Estimates and Mean Square Errors of R with

3,7,3,7,10

221121

====== qpqpnn

56 Journal of Reliability and Statistical Studies, December 2015, Vol. 8(2)

12

|rr

15 20

Reliability Avg.

Estimates

MSE Reliability Avg.

Estimates

MSE

0.021989 0.000582 0.005531 8.08

5

10

−

×

0.016748 0.000526 0.002837 4.82

5

10

−

×

5 0.016616 0.017008 0.000286 0.002792 0.003660 2.44

5

10

−

×

0.019145 0.000324 0.004284 3.04

5

10

−

×

0.184193 0.000310 0.004069 2.83

5

10

−

×

0.019219 0.000355 0.004237 3.30

5

10

−

×

0.098864 0.004131 0.019886 0.000460

0.095907 0.004796 0.014591 0.000423

10 0.098864 0.095491 0.002746 0.001661 0.016505 0.000241

0.102057 0.002773 0.018651 0.0002736

0.099870 0.002757 0.017918 0.000261

0.101985 0.003164 0.018323 0.000297

0.584511 0.012804 0.102787 0.004199

0.580772 0.013095 0.099599 0.004870

15 0.588235 0.593470 0.006776 0.098864 0.095308 0.002635

0.589606 0.006541 0.101879 0.002674

0.590869 0.006622 0.099684 0.002654

0.589494 0.011931 0.100602 0.003813

0.925069 0.002907 0.588104 0.013467

0.929745 0.003305 0.584571 0.013697

20 0.930794 0.930670 0.001782 0.588235 0.589184 0.006215

0.925033 0.001879 0.585346 0.006211

0.926922 0.001842 0.586599 0.006222

0.924809 0.002638 0.578639 0.020662

Table 5b: Average Estimates and Mean Square Errors of

R

with

3,7,3,7,10

221121

====== qpqpnn

Bayesian estimation of reliability in two-parameter … 57

12

|rr

5 10

Reliability Avg.

Estimates

MSE Reliability Avg.

Estimates

MSE

0.527212 0.017147 0.301431 0.012172

0.526890 0.015018 0.310643 0.011810

5 0.526315 0.527249 0.012307 0.310478 0.309978 0.008902

0.526866 0.0126550 0.310687 0.009150

0.526997 0.012590 0.310478 0.009102

0.525530 0.015986 0.274773 0.013459

0.728055 0.011734 0.517795 0.015961

0.718481 0.011729 0.517974 0.014000

10 0.720294 0.721179 0.008835 0.526315 0.524161 0.009955

0.719954 0.009045 0.523790 0.010837

0.720326 0.009006 0.523915 0.010609

0.749458 0.012557 0.519945 0.024380

0.845243 0.007065 0.725654 0.012068

0.838176 0.007881 0.715987 0.012066

15 0.834836 0.826150 0.005976 0.720294 0.708335 0.007628

0.827784 0.0060043 0.7122060 0.008112

0.827390 0.0060046 0.711245 0.007982

0.870079 0.007777 0.756649 0.018033

0.909896 0.003701 0.842101 0.007403

0.907375 0.004407 0.834737 0.007828

20 0.902472 0.876734 0.005034 0.834836 0.793630 0.006973

0.884843 0.004444 0.808061 0.006014

0.882684 0.004592 0.804243 0.006224

0.931743 0.004333 0.873130 0.009625

Table 6a: Average Estimates and Mean Square Errors of

R

with

1,9,1,9,10

221121

====== qpqpnn

58 Journal of Reliability and Statistical Studies, December 2015, Vol. 8(2)

12

|rr

15 20

Reliability Avg.

Estimates

MSE Reliability Avg.

Estimates

MSE

0.176727 0.007721 0.104030 0.004384

0.185015 0.008525 0.107695 0.005208

5 0.183515 0.192788 0.006518 0.108363 0.134013 0.004755

0.190298 0.006536 0.124208 0.004076

0.190934 0.006540 0.126825 0.004242

0.140342 0.009060 0.068923 0.004552

0.299440 0.012136 0.182073 0.008720

0.308754 0.011652 0.190438 0.009657

10 0.310784 0.323621 0.008177 0.183515 0.230966 0.008483

0.318944 0.008760 0.214370 0.007319

0.320141 0.008602 0.218775 0.007569

0.269985 0.020795 0.142069 0.012357

0.528051 0.016868 0.302399 0.012456

0.527508 0.014852 0.311517 0.012054

15 0.526315 0.523521 0.005220 0.310784 0.371186 0.008077

0.523198 0.007006 0.350055 0.007246

0.523318 0.006485 0.355758 0.007328

0.516243 0.038350 0.285457 0.031339

0.728946 0.012284 0.531249 0.015239

0.719389 0.012217 0.530590 0.013301

20 0.720294 0.663570 0.008039 0.526315 0.521966 0.001070

0682566 0.007419 0.522492 0.002797

0.677480 0.007469 0.522351 0.002149

0.740730 0.028737 0.518026 0.055713

Table 7b: Average Estimates and Mean Square Errors of

R

with

1,9,1,9,10

221121

====== qpqpnn