## The Annals of Statistics

### Optimal Rate of Convergence for Finite Mixture Models

Jiahua Chen

#### Abstract

In finite mixture models, we establish the best possible rate of convergence for estimating the mixing distribution. We find that the key for estimating the mixing distribution is the knowledge of the number of components in the mixture. While a $\sqrt n$-consistent rate is achievable when the exact number of components is known, the best possible rate is only $n^{-1/4}$ when it is unknown. Under a strong identifiability condition, it is shown that this rate is reached by some minimum distance estimators. Most commonly used models are found to satisfy the strong identifiability condition.

#### Article information

Source
Ann. Statist., Volume 23, Number 1 (1995), 221-233.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176324464

Digital Object Identifier
doi:10.1214/aos/1176324464

Mathematical Reviews number (MathSciNet)
MR1331665

Zentralblatt MATH identifier
0821.62023

JSTOR