Maximum Likelihood Estimation in Truncated Samples

Abstract

In this paper we consider the problem of estimation of parameters from a sample in which only the first $r$ (of $n$) ordered observations are known. If $r = \lbrack qn \rbrack, 0 < q < 1$, it is shown under mild regularity conditions, for the case of one parameter, that estimation of $\theta$ by maximum likelihood is best in the sense that $\hat{\theta}$, the maximum likelihood estimate of $\theta$, is (a) consistent, (b) asymptotically normally distributed, (c) of minimum variance for large samples. A general expression for the variance of the asymptotic distribution of $\hat{\theta}$ is obtained and small sample estimation is considered for some special choices of frequency function. Results for two or more parameters and their proofs are indicated and a possible extension of these results to more general truncation is suggested.