Maximum Likelihood Estimation in Truncated Samples

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 = qn, 0 < q < 1, it is shown under mild regularity conditions, for the case of one parameter, that estimation of by maximum likelihood is best in the sense that, the maximum likelihood estimate of, 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 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.