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Asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity. (English) Zbl 0637.62026
Summary: 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 \[ \| {\hat \theta}_ n-\theta_ n\| =O_ p(\sqrt{p_ n/n})\quad and\quad \| {\hat \theta}_ n-\theta_ n-\bar X_ n\| =O_ p(p_ n/n), \] where \(\bar X{}_ n\) is the sample mean. The latter result provides normal approximation results when \(p\) \(2_ n/n\to 0\). It is shown by example that even for a single coordinate of (\({\hat \theta}_ n- \theta_ n)\), \(p\) \(2_ n/n\to 0\) may be needed for normal approximation.
However, if \(p_ n^{3/2}/n\to 0\), the likelihood ratio test statistic \(\Lambda\) for a simple hypothesis has a chi-square approximation in the sense that (-2 log \(\Lambda\)-p\({}_ n)/\sqrt{2p_ n}\to_ D\) \({\mathcal N}(0,1)\).

MSC:
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
62F10 Point estimation
62F05 Asymptotic properties of parametric tests
62H12 Estimation in multivariate analysis
60F05 Central limit and other weak theorems
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