INTRODUCTION
Recently a new class of Weighted Exponential (WE) distribution has been proposed
by Gupta and Kundu (2009). Different methods may be
used to introduce a shape parameter to an exponential model and they may result
in a variety of Weighted Exponential (WE) distribution. For example, the gamma
distribution and the generalized exponential distribution are different weighted
versions of the exponential distribution.
The random variable X is said to have a Weighted Exponential (WE) distribution with the shape and scale parameters as α>0 and λ>0, respectively, if it has the following probability density (PDF):
The corresponding Cumulative Distribution Function (CDF) for x>0, becomes:
From now on a WE distribution with the shape and scale parameters as α and λ, respectively will be denoted WE (α, λ).
Note that the WE (α, λ) can be obtained exactly the same way Azzalini
(1985) obtained the skewnormal distribution from two i.i.d. normal distribution.
Suppose X_{1} and X_{2} are i.i.d. exp (λ), i.e., an exponential
random variable with mean 1/λ, then for α>0 consider a new random
variable X = X_{1}, if αX_{1}≥X_{2}. Then X
follows WE (α, λ), Dixit et al. (1996),
assume that a set of random variables (X_{1}, X_{2},..., X_{n})
represent the distance of an infected sampled plant from a 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 know to be carriers of Barley Yellow Mosaic Dwarf Virus
(BYMDV) have passed the virus into the plants when the aphids feed on the sap.
Dixit and Nasiri (2001) considered estimation of parameters
of the exponential distribution in the presence of outliers generated from uniform
distribution. Also new Makhdoom (2011) obtained the
estimation of the parameters of minimax distribution in the presence of one
outlier. In this paper, we obtain the maximum likelihood and moment estimators
of the parameters of the weighted exponential distribution in the presence of
one outlier generated from exponential distribution.
Let the random variable X_{1}, X_{2},..., X_{nk} are
independent, each having the probability density function f_{1} (x):
k random variables (as outlier) are also independent, have the probability
density function f_{2} (x):
The joint density of X_{1}, X_{2},..., X_{n} is given as:
Where:
and:
(Dixit, 1989; Dixit et al.,
1996; Dixit and Nasiri, 2001). The main of this
paper is to focus on obtaining maximum likelihood estimators and moment estimators
of WE (α, λ) with presence of outliers.
JOINT DISTRIBUTION OF (X_{1},..., X_{n}) WITH k OUTLIERS
The joint distribution of X_{1},..., X_{n} in the presence of k outliers after some computations in Eq. 3 is given by:
METHOD OF MOMENT
From Eq. 4, the marginal distribution of X is:
where, x>0, λ>0. Consider,
From Eq. 5 we get:
Let:
and:
From Eq. 7, we have:
MAXIMUM LIKELIHOOD ESTIMATORS OF (α, λ_{1}, λ_{2})
The joint distribution of X_{1},..., X_{n} in the presence of k outliers after some computations in Eq. 3 is given by:
In Eq. 4, if let k = 1(for one outlier), the joint distribution of X_{1},..., X_{n} is given by:
The Loglikelihood function of the observed sample is:
The MLE’s of α, λ_{1} and λ_{2} say
and ,
respectively which is obtained as the solutions of:
There is no closedform solution to this system of equations, so we will solve
for
and
iteratively, using the NewtonRaphson method for root finding. In our case we
will estimate
iteratively:
where, g and G are the vector of normal equations and matrix of second derivatives,
respectively,
The NewtonRaphson algorithm converges, as our estimates of θ, λ_{1}
and λ_{2} change by less than a tolerated amount with each successive
iteration, to
and .
MAXIMUM LIKELIHOOD ESTIMATORS OF (α, λ)
Here, without loss of generality we assume λ_{1} = λ_{2} = λ. Then the NewtonRaphson method be reduced to estimation of θ = (α, λ) in WE (α, λ) with presence one outlier (k = 1) generate from exponential distribution. In this case the loglikelihood function based on the observed sample {x_{1},..., x_{n}} is :
The elements of vector of normal equations (g) and matrix of second derivatives (G) are as following:
The NewtonRaphson algorithm converges, as our estimates of α and λ
change by less than a tolerated amount with each successive iteration, to
and .
Note in case of no outlier presence, Gupta and Kundu (2009)
obtained the estimation of parameters weighted exponential by Maximum likelihood
and Moment method.
NUMERICAL EXPERIMENTS AND DISCUSSIONS
In this study, we have addressed the problem of estimating parameters of Weighted
Exponential distribution in presence of one outlier. In order to have some idea
about Bias and Mean Square Error (MSE) of methods of moment and MLE, we perform
sampling experiments using a SAS. The results are given in Table
1 and 2, when α = 3, λ = 0.5, α = 2 and
λ = 1. We report the average estimates and the MSEs based on 1000 replications.
It is observed that the maximum likelihood estimators are better than the moment
estimators; these are true for all values of n.
Table 1: 
α = 3, λ = 0.5 

Table 2: 
α = 2, λ = 1 

Therefore we suggest using the maximum method for estimating parameters of
the weighted exponential distribution in the presence of one outlier.