## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 23, Number 2 (1952), 226-238.

### 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.

#### Article information

**Source**

Ann. Math. Statist., Volume 23, Number 2 (1952), 226-238.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177729439

**Digital Object Identifier**

doi:10.1214/aoms/1177729439

**Mathematical Reviews number (MathSciNet)**

MR48760

**Zentralblatt MATH identifier**

0047.13302

**JSTOR**

links.jstor.org

#### Citation

Halperin, Max. Maximum Likelihood Estimation in Truncated Samples. Ann. Math. Statist. 23 (1952), no. 2, 226--238. doi:10.1214/aoms/1177729439. https://projecteuclid.org/euclid.aoms/1177729439