Simulation MLE of Parameters of the Mixture Distribution in the Presence of Two Outliers

In the presence paper, we deal with the estimation of parameters of the Exponentiated Gamma (EG) distribution with presence of multiple(r=2) outliers. The maximum likelihood and moment of the estimators are derived. These estimators are compared empirically using Monte Carlo simulation when all the parameters are unknown. There bias and MSE are investigated with help of numerical technique.


Introduction
Dixit, Moore and Barnett (1996), assume that a set of random variables ) , ... , , ( 21 n X X X 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.Deiri (2011,a,b,c) considered the estimation of the parameters of the exponentiated gamma distribution in the presence of outlier generated from exponentiated gamma distribution.In this paper, we obtain the maximum likelihood and moment estimators of the parameters of the exponentiated gamma distribution in the presence of multiple outliers generated from exponentiated gamma distribution.
Recently a new distribution, called Exponentiated Gamma (EG) distribution, has been introduced.This distribution was introduced by Gupta et al. (1998) which has a probability density function (p.d.f.) of the form and a cumulative distribution function (c.d.f.) where  is the shape parameter.It is important to mention that when 1   , the Exponentiated Gamma p.d.f. is that of gamma distribution with shape parameter 2   and scale parameter .For more details about this distribution, see Shawky andBakoban (2008c &2009).Also, characterization from EG distribution based on record values and Bayesian estimations on the EG distribution discussed by Shawky and Bakoban (2008b,a).The density functions of the Exponentiated Gamma distribution can take different shapes.Figure 1 shows the shape of  We assume that the random variables are such that two of them are distributed with p.d.f in the presence of two outliers.Section 3 and 4 discusses the methods of moment and maximum likelihood estimators.The different proposed methods have been compared using Monte Carlo simulations and the results have been reported in section 5.

Joint
x e e x x e e x c x e e x where Dixit (1996), Dixit and Nasiri (2001), and Deiri (2011a,b,c).From ( 4), the marginal distribution of X is

Method of Moment
The moments can also be obtained in the form of a series which is finite or infinite depending on whether  and  are integers or not.The raw moments of X may be determined from (5) be direct integration.For   r , we find that x e e x dx n n x e e x dx n n for 0  x by using the binomial series expansion we have Also, whereas i is integer, by using the binomial series expansion we have Since the quantity inside the summation is absolutely integrable, interchanging the summation and integration we have ( 1) ( 1) in case of no outlier presence, from (9) we get it is proposed by Gupta et al. (1998), also in case of two outlier presence, it is given by Deiri (2011Deiri ( , 2012)).We observe that the infinite series is summable.For r = 1, 2 and 3, ) ( r X E is given by ) 00 ( 1)

Method of Maximum Likelihood
One sees from (6) that moment estimates for the parameters of the EG distribution with presence of two outliers can not be obtained in closed forms and therefore that is little point in considering the method any further.Proceeding with the method of maximum likelihood, the log likelihood function from a sample of n observations, ( 2) ln 2ln ln ln ( 1) ln 1 (1 ) ln 1 (1 ) where Taking the derivative with respect to  and  and equating to 0, we obtain the normal equations as (1 ) (1 ) Here, we need to use either the scoring algorithm or the Newton-Raphson algorithm to solve the two non-linear equations simultaneously, so we will solve for  ˆ and  ˆ iteratively, using the Newton-Raphson method, a tangent method for root finding.In our case we will estimate ) , where g is the vector of normal equations for which we want and G is the matrix of second derivatives (1 ) (1 ) ln 1 (1 )

()
(1 ) ) The Newton-Raphson algorithm converges, as our estimates of  and  change by less than a tolerated amount with each successive iteration, to ˆ and  ˆ.

Note that for
, in case of no outlier presence,  ˆ can be obtain as it is given by Gupta et al. (1998), also in case of two outlier presence, it is obtained by Deiri (2011).

Numerical Experiments and Discussions
In this paper, we have addressed the problem of estimating parameters of Exponentiated Gamma distribution in presence of two outliers.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 MATLAB.The results are given in Tables 1 and 2, for 0.5   and 2   .We report the average estimates and the MSEs based on 1500 replications.It is observed that the maximum likelihood estimator work quit well.
of  .