Differentiability of inverse operators and limit theorems for inverse functions
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. Wahrscheinlichkeitstheorie Verw. Geb. 23, 245-253 (1972; Zbl 0238.60018)] to the multivariate case. Each of these equivalent statements implies a kind of Taylor expansion of the SC-inverse \(G^{-1}_n\) at \(G\) (locally uniformly in \(T\)) \[ G^{-1}_n=G^{-1}-\text{inv}(G'\circ G^{-1})(G_n-G)\circ G^{-1}+o_P(a_n^{-1}) \;\;\text{as} \;n\to\infty, \] where \(\text{inv}(A)\) denotes the inverse of a nonsingular linear transformation \(A\) in \(R^d.\) Such limit theorems for functional ``inverses'' can be used to study asymptotic behavior of statistical estimators defined implicitly (as solutions of equations involving the empirical distribution \(P_n\)). It is shown how to apply this approach to get asymptotic normality of \(M\)-estimators in the multivariate case under minimal assumptions. An extension of the quantile function to the multivariate case related to \(M\)-parameters of a distribution \(P\) in \(R^d\) (an \(M\)-quantile function) is considered. To study limit behavior of the empirical \(M\)-quantile process limit theorems for functional ``inverses'' are used. Asymptotics of regression quantiles are also studied.
scientific article; zbMATH DE number 1208777