Estimation of reliability parameter
based on censored samples
Michael S. Tikhov
Dept of Applied Mathematics
Lobachevsky State University
Nizhny Novgorod, 603600, av. Gagarin, 23
Russia
[email protected]
Abstract
Let X 1 , X 2 , ... , X n are independent and identically distributed random variables with distribution
function given by F ( x ;  )  1  exp ( ( x   )), x   ,   0 . We observe only the order statistics
xn(k 1,nm)  ( xn(k 1) , xn(k 2) , ..., xn(nm) ) . The smallest k and the largest m values are censored. This
report analyzes the problem of estimating of the function R( x   )  exp ( ( x   )) , x   for given
value x based on the failure time data xn(k 1,nm) . In addition we consider the estimation of the
reliability parameter from a random censored samples.
1 Introduction
Estimation of the reliability parameter is a major problem of the statistical reliability theory.
However the majority of the used methods of estimation is based on the complete samples and
assumption of both mutual independence and identical distributed of the outcomes of observations,
while the data can be of type II censored or random censored. It creates difficulties in processing and
interpretation of such failure data. In the present report we shall offer estimators of distribution
parameters on the incomplete samples.
Let us assume that it is possible to treat the failure data as the independent, identically distributed
as X random variables (rv) X 1 , X 2 , ... , X n with the distribution function (df) F ( x ; ) , dependent
on unknown parameter  , x n(i ) , 1  i  n , is the order statistics, constructed on the sample
x 1 , x2 , ... , xn . The problem is to determine of suitable estimators and function reliability
R ( x ;  )  1  F ( x ; ) .
Let us consider the problem of the estimation of function R( x   )  exp ( ( x   )) , x   ,  is
unknown, by outcomes of the observation of a set the order statistics
x n( k 1,n  m)  ( x n( k 1) , ... , x n( n  m) ) . If the sample is complete, then as unbiased estimator for R ( x   )
usually the statistics Rˆ ( x)  (1  1 / n) exp ( ( x  t )) is taken, if x  t , and Rˆ ( x)  1 , if x  t , where
t  x n(1) (see Nikulin, Voinov (1989), p.348). The same statistics is the unbiased estimator of
R ( x   ) also in the case k  0 , m  0 . It is necessary to note, that the function R ( x   ) is a
continuous function, while the estimator Rˆ ( x) is a discontinuous function (in the point x  t it has a
jump of magnitude 1 / n . We propose the Bayes estimator for the function R ( x   ) in the point x
on the sample x n( k 1,n m) .
2 Censored data
For improper density of prior distribution, say  ( )  exp (   ) ,       , it has the form




t n( k ) ( x)  




(1  k / n) e (t  x ) ,
e ( x t )( n  k )

if x  t ;
( n)
Г (n  k ) Г (k )
1


F ( n  k  1 , 1  k ; n  k  2 ; e x t ) 
 n  k 1
1

F (n  k , 1  k ; n  k  1; e x t )  , if x  t ,
nk

where t  x n( k ) ,  (n)  (n  1) ! and F ( a , b ; c ; x ) is Gauss hipergeometric function:
ab
1 a (a  1) b (b  1) 2
F ( a , b; c; x ) 1
x
x  ... , c  0 ,  1,  2 , ... .
c
2!
c (c  1)
For the complete sample x n(1,n ) it has form:
t n(1) ( x)
 (1  1 / n) e t  x , if x  t ;

 1  (1 / n) e ( x t )(n 1) , if x  t .
The estimator t n( k ) ( x) will be Bayesian estimator on the sample x n( k 1,n m) with replacement, in the
case x  t , in the right part n by n  m . The estimator t n( k ) ( x) can be rewritten in such a form:
t n( k ) ( x)  1 
 (n)  e ( x t )(nk )
 (n  k )  (k )

C k1 e x t
1



 (n  k  1) (n  k ) (n  k  2) (n  k  1)

C k2 e 2 ( x t )
(1) k e ( k 1) ( x t )

 ... 
(n  k  3) (n  k  2)
n (n  1)

 , if x  t .


If the density of prior distribution is equal exp (   ) , when   0 , and is equal to 0, when   0 ,
then Bayes estimator for R ( x   ) is
 e t  x (1  1 / n) (1  e  n t ) (1  e  ( n1) t ) 1 , if

tˆn(1) ( x)  
x t;
n  1  x  ( n1) t 1 ( x t ) ( n1) 
 ( n 1) t 1 
)  1
e
 e
 , if x  t .
 (1  e
n
n



The offered estimators are continuous function of x .
Example. Let 0.05, 0.09, 0.11, 0.15, 0.22, 0.27, 0.28, 0.31, 0.40, 0.65, 0.72, 0.73, 0.79, 0.90, 0.94,
0.99, 1.04, 1.05, 1.23, 1.29, 1.31, 1.37, 1.74, 2.41, 4.26 be the outcomes of 25 observations of a
quantity having an exponential distribution E(1) with df F ( x ; )  1  e  x , x  0 .
(1)
( 2)
(5)
For considered example x25
 0.05 , x25
 0.09 , x25
 0.22 .
In the table 1 we present the estimates of function R ( x   ) for various k and n .
Table 1. The values of the estimates of function R ( x ) .
x
.01
.02
.03
.04
.05
.1
.2
.3
.4
.5
e
R25 ( x)
.990
1.0
.980
1.0
.970
1.0
.961
1.0
.951
.960
.905
.913
.819
.826
.741
.748
.670
.677
.607
.612
(1)
t 25
( x)
.985
.980
.975
.968
.960
.913
.826
.748
.677
.612
.976
.972
.966
.960
.954
.911
.824
.746
.675
.610
.955
.950
.945
.939
.933
.898
.816
.738
.668
.604
.999
.997
.994
.988
.980
.932
.843
.764
.691
.625
x
( 2)
t 25
( x)
(5)
t 25
( x)
(1)
tˆ25
( x)
x
.6
.7
.8
.9
1.0
1.5
2.0
2.5
3.0
4.0
e x
R25 ( x)
.549
.554
.497
.501
.449
.453
.407
.410
.368
.371
.223
.225
.135
.137
.082
.083
.050
.050
.018
.018
(1)
t 25
( x)
.554
.501
.453
.410
.371
.225
.137
.083
.050
.018
( 2)
t 25
( x)
.552
.500
.452
.409
.370
.225
.136
.083
.050
.018
(5)
t 25
( x)
.547
.495
.448
.405
.367
.222
.135
.082
.050
.018
(1)
tˆ25
( x)
.565
.512
.462
.419
.379
.229
.140
.085
.051
.018
Let X 1 , ... , X n be independently distributed according to uniform distribution U ( 0 , ) / In
Nikulin, Voinov (1989), p.284, 285, are given the unbiased estimations of parameter  of this
distribution and its functions. Unlike Nikulin, Voinov (1989) we consider the censored sample, i.e.
we assume, that a set the order statistics x n( k ,n m) , 1  i  n  m  n , is observed.
Here the unbiased estimation of   , where   m  n , is the statistics
 ( n  m )  (  n  1) 
ˆ 
T , T  x n( n m) ,
 ( n  1 )  (  n  m )
and the unbiased estimation of the variance D ( ˆ ) is equal
2


ˆ (ˆ )   (n  m )  ( 2  n  1 )    ( n  m )  (  n  1)  ( 2  n  m ) 1  T 2 .
D

 ( n  1 )  (  n  m )   ( n  1 )  2 (  n  m )  ( 2  n  1)

In particular,
ˆ
n 1
n  m 1

T.
, m  n  1;
ˆ 1 
2 2(n  m)
nT
Also the unbiased estimation for ln  is equal
ln̂   ln T 
m

j 0
1
.
nm j
The unbiased estimation for ln 2  is equal
 m
  m
2
1
1
 
 
ln̂ 2   ln 2 T  2 ln T   
 j 0 n  m  j   j 0 n  m  j 

 

m
1
j 0
( n  m  j )2

.
3 Random censoring
Let us consider now the estimation of the parameter  of distribution F ( x ;  ) from the random
censored sample ( Z i , Wi ) , 1  i  n , where rv Z  min ( X 1 , X 2 , ... , X m , Y1 , ... , Yk ) , and W  1 , if
rv X *  min ( X 1 , X 2 , ... , X m ) is observed, and W  0 , if – one of Y . It is supposed, that
X 1 , X 2 , ... , X m are independent and have identical distribution F ( x ;  ) with density function
f ( x ;  ) , and Yi , 1  i  k , are independent and are distributed each with distribution function
Gi ( y ) , both dependent or independent from  . For construction of an estimator of the parameter 
in this scheme it is necessary to make the likelihood function
n
L (  )   ( f ( z j ; ) ( 1  F ( z j ; )) m 1 )
j 1
wj
( 1  F ( z j ; ))
m (1 w j )
.
The MLE of  is that value ˆn , which maximizes the likelihood function L (  ) , i.e.
L ( ˆn )  max L (  ) .
(1)

If m  1 , k  1 , i.e. Z  min ( X , Y ) , W  I ( Z  X ) , then the sample ( z1 , w1 ) , ( z 2 , w2 ) , … ,
( z n , wn ) is called random censored. Let us assume that rv X has df F ( x ;  ) and the density
f ( x ; ) , and rv Y - df G ( x ) . Sown that the MLE ˆ , determined ( 1 ), are asymptotic unbiased
n
and asymptotic normal (see Tikhov (1991)).
References
Nikulin, M.S., Voinov, V.G. (1989). Unbiased estimates and its applications. M.: Nauka.
Tikhov, M.S. (1991) Bayes type estimates. J. Soviet. Math. 56, 2438-2442.