Estimating the Parameters of Truncated Distributions

The problem considered here is the estimation of the parameters of some special truncated distributions. If the sample observations are restricted to the interval {O, T} with T known, then it is well known in the literature that the method of maximum likelihood fails to provide a finite estimate, for the mean of an exponential distribution, whenever the sample mean is greater than T/2 (Deemer and Votaw, 1955, Ann. Math. Statist. 26, 498-504). Not so well known is the nonexistence of the maximum likelihood estimator (m.l.e.), under certain conditions, for the scale parameter of a gamma distribution from a truncated sample, when the shape parameter of the distribution is assumed known (Broeder, 1955, Ann. Math. Statist. 26, 659-663). The above-mentioned results do not hold when the sample observations are truncated to an infinite interval, say to {T,(INFIN)), in which case the m.l.e. exists with probability one. This research deals with similar results pertaining to the estimation of mean and standard deviation of a normal distribution from a doubly truncated sample, such that the sample observations are within the interval {A,B}, - (INFIN) \u3c A \u3c B \u3c (INFIN), A and B known. It is proved here that the m.l.e.\u27s (which are the same as the moment estimators) of these parameters are nonexistent with positive probability. The cases for the two-parameter gamma and Weibull distributions are also examined with the help of Broeder\u27s technique of standardizing the truncation interval to {0,1} through a simple transformation. In the cases considered here, the m.l.e\u27s even when they exist, exhibit a tendency of blowing up near the upper boundary of the interval of their existence. In order to correct this problem, as well as to find estimators that exist with probability one, the class of Bayes modal, or modified maximum likelihood estimators is considered. The Bayes modal estimators were introduced by Blumenthal and Marcus (1975, J. Amer. Statist. Assoc. 70, 913-922). A new estimation procedure combining the m.l.e. and the Bayes modal estimator, called the mixed estimator, is proposed here. Simulations provide the comparison of the aggregate behavior of the m.l.e.\u27s, the modal estimators, and the mixed estimators