The Annals of Statistics

Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity

Stephen Portnoy

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Abstract

Consider a sample of size $n$ from a regular exponential family in $p_n$ dimensions. Let $\hat\theta_n$ denote the maximum likelihood estimator, and consider the case where $p_n$ tends to infinity with $n$ and where $\{\theta_n\}$ is a sequence of parameter values in $R^{p_n}$. Moment conditions are provided under which $\|\hat\theta_n - \theta_n\| = O_p(\sqrt{p_n/n})$ and $\|\hat\theta_n - \theta_n - \overline{X}_n\| = O_p (p_n/n)$, where $\overline{X}_n$ is the sample mean. The latter result provides normal approximation results when $p^2_n/n \rightarrow 0$. It is shown by example that even for a single coordinate of $(\hat\theta_n - \theta_n), p^2_n/n \rightarrow 0$ may be needed for normal approximation. However, if $p^{3/2}_n/n \rightarrow 0$, the likelihood ratio test statistic $\Lambda$ for a simple hypothesis has a chi-square approximation in the sense that $(-2 \log \Lambda - p_n)/\sqrt{2p_n} \rightarrow_D \mathscr{N}(0, 1)$.

Article information

Source
Ann. Statist., Volume 16, Number 1 (1988), 356-366.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350710

Digital Object Identifier
doi:10.1214/aos/1176350710

Mathematical Reviews number (MathSciNet)
MR924876

Zentralblatt MATH identifier
0637.62026

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems 62F10: Point estimation

Keywords
Asymptotics maximum likelihood central limit theorem exponential family

Citation

Portnoy, Stephen. Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity. Ann. Statist. 16 (1988), no. 1, 356--366. doi:10.1214/aos/1176350710. https://projecteuclid.org/euclid.aos/1176350710


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