March, 1988 Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity
Stephen Portnoy
Ann. Statist. 16(1): 356-366 (March, 1988). DOI: 10.1214/aos/1176350710
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 $(-2 \log \Lambda - p_n)/\sqrt{2p_n} \rightarrow_D \mathscr{N}(0, 1)$.
Stephen Portnoy. "Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity." Ann. Statist. 16 (1) 356 - 366, March, 1988. https://doi.org/10.1214/aos/1176350710
Published: March, 1988
First available in Project Euclid: 12 April 2007
MathSciNet: MR924876
Primary: 62E20
Secondary: 60F05, 62F10
Keywords: asymptotics, central limit theorem, exponential family, maximum likelihood
Vol.16 • No. 1 • March, 1988
Stephen Portnoy "Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity," The Annals of Statistics, Ann. Statist. 16(1), 356-366, (March, 1988)