Parameter estimation in smooth empirical processes
type: Item
instance of: scholarly article
title: Parameter estimation in smooth empirical processes
zbMATH Open document ID: 0612.62040
DOI: 10.1016/0304-4149(86)90003-7
published in: Stochastic Processes and their Applications
publication date: 1986-01-01T00:00:00Z
review text: 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 representati…
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 3990617
zbMATH Keywords: M-estimates
zbMATH Keywords: R-estimates
zbMATH Keywords: L-estimates
zbMATH Keywords: generalized likelihood equation
zbMATH Keywords: tightness
zbMATH Keywords: minimum distance estimates
zbMATH Keywords: Riesz derivatives
zbMATH Keywords: limit distributions
zbMATH Keywords: L-functions of integral type
zbMATH Keywords: empirical process
zbMATH Keywords: bounds
zbMATH Keywords: oscillation modulus
zbMATH Keywords: almost sure representation
zbMATH Keywords: sequential fixed width confidence intervals
zbMATH Keywords: Banach space approach
zbMATH Keywords: asymptotic normality
author: Winfried Stute
MaRDI profile type: Publication
full work available at URL: 0304-4149(86)90003-7
OpenAlex ID: W1978887517
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Wikidata QID: Q127216894
description: scientific article; zbMATH DE number 3990617