## The Annals of Statistics

- Ann. Statist.
- Volume 1, Number 5 (1973), 888-901.

### On the Measurability and Consistency of Maximum Likelihood Estimates for Unimodal Densities

#### Abstract

This paper is concerned with maximum likelihood estimates for a large class of families of unimodal densities. The existence of measurable maximum likelihood estimates and the consistency of asymptotic maximum likelihood estimates are proved. By counterexamples it is shown that the conditions which are sufficient for consistency cannot be removed without compensation.

#### Article information

**Source**

Ann. Statist., Volume 1, Number 5 (1973), 888-901.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342509

**Digital Object Identifier**

doi:10.1214/aos/1176342509

**Mathematical Reviews number (MathSciNet)**

MR368286

**Zentralblatt MATH identifier**

0274.62028

**JSTOR**

links.jstor.org

**Subjects**

Primary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Secondary: 62G05: Estimation 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54E45: Compact (locally compact) metric spaces

**Keywords**

Unimodal density the mode of a unimodal density upper and lower semicontinuity Levy metric convergence in the mean pointwise convergence compact and locally compact metric space weak topoloty and topology induced by the supremum-metric on families of probability measures asymptotic maximum likelihood estimates existence of measurable estimates strong consistency

#### Citation

Reiss, Rolf-Dieter. On the Measurability and Consistency of Maximum Likelihood Estimates for Unimodal Densities. Ann. Statist. 1 (1973), no. 5, 888--901. doi:10.1214/aos/1176342509. https://projecteuclid.org/euclid.aos/1176342509