Parameter estimation in smooth empirical processes
Let \(\{F_{\theta}:\theta\in \Theta \}\), \(\Theta\subset {\mathbb{R}}\), be some family of distribution functions (d.f.) on the real line, where \(F_{\theta}\) is not necessarily assumed to be completely specified by \(\theta\), the parameter of interest. Let \(\xi_ i\), \(i\geq 1\), be i.i.d. with d.f. F, where F is equal or at least ''close'' to some \(F_{\theta_ 0}\), \(\theta_ 0\in \Theta\). A general method for estimating \(\theta\) is to minimize (or maximize), for a given sample size n, a parametric function of the emprical d.f. \(F_ n\) pertaining to \(\xi_ 1,...,\xi_ n\). Under appropriate smoothness assumptions this problem may be reduced to finding the solution \(\theta_ n\) of a generalized (log-) likelihood equation \(L(\theta,F_ n)=0\), where \(L(\theta_ 0,F)=0.\) This paper is concerned with the important class of L-functions of integral type, i.e. for a d.f. G we have \(L(\theta,G)=\int T(x,\theta,G)G(dx)\), where T is assumed to be a smooth function of \(\theta\), whereas T may depend on G in a quite arbitrary way which yields the wide scope of applicability of the presented results. An important role is played here by the empirical process \[ \alpha_ n(\theta):=n^{1/2}\int T(x,\theta,F)[F_ n(dx)-F(dx)],\quad \theta \in \Theta_ 0, \] \(\Theta_ 0\subset \Theta\) compact, associated with the family of functions \(\{T(\cdot,\theta,F):\theta \in \Theta_ 0\}\). Under a smoothness condition on T (as a function of \(\theta)\), this process will be a random element in the Banach space of all real continuous functions on \(\Theta_ 0.\) Conditions are investigated under which \(\{\alpha_ n:n\in {\mathbb{N}}\}\) is tight and explicit bounds for the oscillation modulus of \(\alpha_ n\) are proved. Under a monotonicity assumption on \(\theta\mapsto T(x,\theta,G)\) LIL-upper bounds are derived for the deviation between \(\theta_ n\) and \(\theta_ 0\). Together with the estimate for the oscillation modulus of \(\alpha_ n\), this yields an almost sure representation of \(\alpha_ n\) being useful for deriving sequential fixed width confidence intervals for \(\theta_ 0\). Based on the above mentioned Banach space approach and some well known facts about empirical processes, a relatively self-contained proof of the asymptotic normality of \(\theta_ n\) is presented. (From the author's introduction.)
sequential fixed width confidence intervals
scientific article; zbMATH DE number 3990617
scientific article