Estimation of Parameter in a New Truncated Distribution

This paper discusses the estimation of the parameter in a truncated form of a discrete distribution which is analogous to Burr distribution. The maximum likelihood and the moment estimators of the parameter are obtained. Their asymptotic properties are also established.


Introduction
Sreehari [1] has derived a class of discrete distributions analogous to Burr family by solving a differential equation.The probability mass function (pmf) of the random variable X having the d-th class of the distributions derived by Sreehari (2010) is with 0 <  < 1.
Nanjundan and Naika [2] have discussed the estimation of the parameter of this distribution.
When this distribution is truncated at 0, the pmf of X turns out to be ( 1)! 0, otherwise.
We get It is straightforward to observe that E(X) > Var(X).That is this truncated distribution is under dispersed.
The Fisher information measure is computed in Section 2. The maximum likelihood (ML) estimation of θ is discussed in Section 3 whereas the method of moment estimate of θ is obtained in Section 4.An asymptotic comparison of the maximum likelihood and method of moment estimates is done in Section 5.The results of a simulation study are presented in Section 6.

Fisher Information Measure
When X has the pmf specified in (2), we get The Fisher information measure corresponding to this pmf is given by is not tractable, it can be numerically evaluated for the required values of θ.
! is convergent and it can be evaluated numerically.

Maximum Likelihood Estimation
be a random sample on X having the pmf specified in (2).Then the likelihood corresponding to the sample is The maximum likelihood (ML) estimator is given by and hence the ML estimator of θis the solution of Since (3) does not yield a closed form expression for the ML estimator, it can be evaluated by a numerical procedure like Newton-Raphson method.Let ˆmle  denote the ML estimator of θ.
The pmf in (2) satisfies the following regularity conditions.
1) The support of X does not depend on the parameter θ.
is twice differentiable w.r.t.θ under the summation sign.
5) There exists a function such that The conditions 1)-5) are easy to verify.We prove the validity of 5). Here where 0  is the true value of the parameter.Since the parameter space is an open interval such a neighborhood exists.Obviously, p(x, θ) satisfies all the regularity conditions of Cramer [3] and it belongs to Cramer family.Hence In other words, ˆmle  is consistent and asymptotically normal (CAN) for θ.

Method of Moment Estimation
 be a random sample on X having the pmf specified in (1.2) and the method of moment (MM) estimator of θ is given by Evidently the above equation does not admit a closed form expression of the MM estimator of θ and hence for Copyright © 2013 SciRes.

OJS
an observed sample, it has to be numerically computed.Let ˆmme  denote the moment estimator of θ.Since p(x, θ) does not belong to exponential family, the asymptotic normality of ˆmme  is not automatic.We establish this property using the delta method.
Note that 1 2 , , , n X X X  are independent and identically distributed (iid) with is the moment estimate of θ.By Levy-Lindeberg central limit theorem, and is continuous.
Hence by the delta method (see Knight [4]),  is also CAN for θ.

Asymptotic Relative Efficiency
The asymptotic relative efficiency of the ML estimator over the method of moment estimator is given by  has no closed form expression and is computed numerically, the ARE has also to be numerically evaluated.The Table 1 shows the ARE for various values of θ.
The ARE of the ML estimator over the MM estimator is steadily increasing as θ increases.That is the ML estimator uniformly performs better than the MM estimator.

Simulation Study
A modest simulation study has been carried out using the R software.One thousand samples, of size 100, 250, and 500 were generated from the truncated distribution specified in (2) for θ = 0.5.Both the ML and the MM estimates were computed solving respectively (3) and ( 4) using Newton-Raphson method.Sample means were taken to be the initial estimates.Since the asymptotic normality of both estimates has been analytically established, an elaborate simulation study has not been done.However the following histograms give graphical evidence of the asymptotic normality of the estimates.As the sample size increases, the estimates tend to be distributed more normally.
The Figures 1-3 show the histograms of the MLEs and the MMEs based on 1000 samples of sizes 100, 250, and 500 and θ = 0.5.The distribution discussed in this paper is structurally similar to the truncated Poisson distribution and under dispersed.Hence this distribution can be used as an alternative to the truncated Poisson when the data exhibit under dispersion.Both the ML and MM estimators do not have closed form expressions.But they can easily be computed using Newton-Raphson method.Both of the estimators are CAN for the parameter.The ML estimator is asymptotically more efficient than the MM estimator.


Since the asymptotic variance of ˆmle

Figure 1 .
Figure 1.The histograms of the MLEs and the MMEs for sample size 100.