Differentiability of inverse operators and limit theorems for inverse functions
type: Item
instance of: scholarly article
title: Differentiability of inverse operators and limit theorems for inverse functions
zbMATH Open document ID: 0916.60001
DOI: 10.1023/A:1022650530662
published in: Journal of Theoretical Probability
publication date: 1999-07-12T00:00:00Z
review text: Let \(H\) be a map from a set \(S\subset R^d\) to \(R^d.\) The following definition of the inverse map \(H^{-1}\) is considered. For \(t\in R^d\) let \(\delta_H(t)\) denote the Euclidean distance from \(t\) to the set \(H(S).\) Consider sequences \(\{s_n\}_{n\geq 1}\) in \(S\) such that \(| t-H(s_n)| \to \delta_H(t)\), \(n\to\infty.\) Any limit point of any such sequence (finite or infinite) is considered as a possible value of the ``inverse'' \(H^{-1}(t).\) Any map \(t\to H^{-1}(t)\) defined in such a way is called an SC-inverse (a selected closest inverse) to \(H\). Differentiability of the nonlinear operator \(H\to H^{-1}\) at \(H=G,\) where \(G\) is a one-to-one map from \(S\) onto a set \(T\subset R^d\) with good analytic properties (specially, a diffeomorphism), is studied. Compact differentiability of this operator tangentially to continuous functions is established. A family of norms is defined such that it is Frechet differentiable with respect to them and optimal bounds for the remainder of the differentiation is obtained. This extends recent results of \textit{R. M. Dudley} [Ann. Stat. 22, No. 1, 1-20 (1994; Zbl 0816.62039)] to the multivariate case. These differentiability results are applied to random maps \(G_n: S\to R^d,\) which could be statistical estimators of an unknown map \(G.\) For a function \(J\) on \(R^d,\) let \((J)_T\) be its restriction to \(T.\) It is shown that for a diffeomorphism \(G\) and for an increasing sequence of positive numbers \(\{a_n\}_{n\geq 1}\) weak convergence of the sequence \(\{a_n(G_n-G)\}_{n\geq 1}\) (locally in \(S\)) is equivalent to weak convergence of the sequence \(\{a_n(G^{-1}_n-G^{-1})\}_{n\geq 1}\) (locally in \(T\)) along with the convergence of the sequence \(\{a_n((G^{-1}_n)^{-1}_T-G_n)\}_{n\geq 1}\) to \(0\) in probability (locally uniformly in \(S\)). The equivalence holds for all SC-inverses \(G^{-1}_n\) and all double SC-inverses \((G_n^{-1})^{-1}_T\) and it extends a theorem of \textit{W. Vervaat} [Z.…
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 1208777
zbMATH Keywords: weak convergence
zbMATH Keywords: quantile processes
zbMATH Keywords: regression
zbMATH Keywords: empirical distribution
author: Vladimir Koltchinskii
MaRDI profile type: Publication
full work available at URL: a:1022650530662
OpenAlex ID: W280864121
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description: scientific article; zbMATH DE number 1208777