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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