Parviz Nasiri et al./ Elixir Statistics 52A (2012) 11691-11695 11691
Introduction
The exponential distribution plays an important role in life
testing problems. A great deal with research has been done on
estimating the parameters of exponential distribution using both
classical and Bayesian techniques and a very good summary of
this work can be found in Johanson, Kotz, and Balakrishnan
(1994). There are also some papers on estimation and prediction
for exponential distribution parameters based of record and
censored samples. See for example Balasubramanian and
Balakrishnan (1992), Ohandrasekar, Leo Alexander and
Balakrishnan (2002), Jaheen (2004), Ahmadi, Doostparast and
Parsian (2005) and references therein. In 2001, Dixit and Nasiri
have estimated the parameter of exponential distribution with
presence of outliers, and in 2009 Asgharzadeh has estimated the
parameter of exponential model based on records value.
In section two, we estimate Bayes estimator of parameter of
the exponential distribution with presence of outlier, in section
three, we estimate the parameter
( )
θ
of exponential distribution
based on records value with presence of outliers, in section four,
we estimate , Bayes estimation of
( )
θ
based on records value
with presence of outliers. These estimators compare in section
five.
Bayes estimation of
θ
with presence of outliers
According to Dixit, Moor and Barnett (1996), we assume
that a set of random variables
( )
n
X X X ,..., ,
2 1
represent of
the distribution of an infected sampled plant from a plot of
plants inoculated with a virus. Some of the observations are
derived from the airborne dispersal of the spores and are
distributed according to the exponential distribution. The other
observations out of n random variables (say
k
) are present
because aphids which are known to be carriers of BYMDV have
passed the virus into the plant when the aphids feed on the sap.
Theses
k
(known) aphids are considered to be exponential
distribution. Thus, we assume that the random variables
( )
n
X X X ,..., ,
2 1
are such that
k
of them are distributed
with probability density function (pdf)
( )
θ λ , , x g
as
( ) > > >
- = θ λ
θ
λ
θ
λ
θ λ , , , exp , x x x g
(1)
and the remaining
) ( k n -
random variables are distributed
with the following pdf.
( ) > >
- = θ
θ θ
θ , , exp
1
x
x
x f
(2)
Then the joint pdf of
( )
n
X X X , ,
2 1
is
( )
( )
( )
( )
( )
∏ ∏
*
= =
-
=
k
j A
A
n
i
i
j
j
x f
x g
x f
n
k n k
x f
1 1
!
! !
, θ θ λ
( )
∏
*
=
=
-
-
-
-
=
k
j A
xA
n
i
i
n
j
j
x
x
n
k n k
1
1
exp
1
exp
exp
1
!
! !
θ θ
θ
λ
θ
λ
θ θ
(3)
( )
- -
-
-
=
=
* k
j
A
k
j
x
n
x n
n
k n k
1
1
exp
exp
!
! !
θ θ
λ
θ
θ
λ
(4)
where
+ =
+ -
+ =
+ -
=
*
-
=
1
2
1
1
1 1 2 1 1 k k
A A
k n
A A
k n
A A
(5)
From (3) the marginal distribution of
X
is given by
()
( )
>
- +
-
+
- = x
x
n
k n x
n
k
x f , exp
1
exp
θ θ θ
λ
θ
λ (6)
Tele:
E-mail addresses: jabbarinm@yahoo.com, jabbarinm@um.ac.ir
© 2012 Elixir All rights reserved
On Bayesian estimator from exponential distribution based on records with
presence of outliers
Parviz Nasiri
1
and Mehdi Jabbari Nooghabi
2
1
Department of Statistics, University of Payame Noor, 19395-4697 Tehran, I. R of Iran.
2
Department of Statistics, Ferdowsi University of Mashhad, Mashhad, I. R of Iran.
ABSTRACT
In this paper, Bayes estimator is derived for the parameter of the exponential model with
presence of outliers based on records value. This estimator compared with estimator when
we not use of records value.
© 2012 Elixir All rights reserved.
ARTICLE INFO
Article history:
Received: 24 September 2012;
Received in revised form:
19 November 2012;
Accepted: 29 November 2012;
Keywords
Exponential model,
Bayes estimators,
Outliers,
Records value.
Elixir Statistics 52A (2012) 11691-11695
Statistics
Available online at www.elixirpublishers.com (Elixir International Journal)