## The Annals of Statistics

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

Stephen Portnoy

#### 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

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