Differentiability of Inverse Operators and Limit Theorems for Inverse Functions

Let H be a map from a set S⊂R d to R d . For tεR d let δ H (t) denote the distance from t to the set H(S). Consider sequences {s n} n≥1 in S such that $$|t - H(s_n )| \to \delta _H (t),{\text{ }}n \to \infty$$ . Any limit point of any such sequence (finite or infinite) is considered as a possible value of the “inverse” H −1(t). Any map $$t \mapsto H^{ - 1} (t)$$ defined in such a way will be called an SC-inverse (a selected closest inverse) to H. In the paper we study differentiability of the nonlinear operator $$H \mapsto H^{ - 1} $$ at H=G, where G is a one-to-one map from S onto a set T⊂R d with good analytic properties (specifically, a diffeomorphism). We establish compact differentiability of this operator tangentially to continuous functions and introduce a family of norms such that it is Fréchet differentiable with respect to them. We also obtain optimal bounds for the remainder of the differentiation, extending to the multivariate case recent results of Dudley. These differentiability results are applied to random maps $$G_n :S \mapsto R^d$$ , 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≥1 weak convergence of the sequence {a n (G n − G)} n≥1 (locally in S) is equivalent to weak convergence of the sequence $$\left\{ {a_n (G_n - G)} \right\}_{n \geqslant 1}$$ (locally in T) along with the convergence of the sequence $$\left\{ {a_n ((G_n^{ - 1} )_T^{ - 1} - G_n )} \right\}_{n \geqslant 1}$$ to 0 in probability (locally uniformly in S). The equivalence holds for all SC-inverses $$G_n^{ - 1}$$ and all double SC-inverses $$(G_n^{ - 1} )_T^{ - 1}$$ and it extends to the multivariate case a theorem of Vervaat. Moreover, each of these equivalent statements implies a kind of Taylor expansion of the SC-inverse $$G_n^{ - 1}$$ at G (locally uniformly in T) $$G_n^{ - 1} = G^{ - 1} - {\text{inv(}}G\prime \circ G^{ - 1} )(G_n - G) \circ G^{ - 1} + o_P (a_n^{ - 1} ){\text{ as }}n \to \infty $$ where 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 ). We show how to apply this approach to get asymptotic normality of M-estimators in the multivariate case under minimal assumptions. We consider 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)and use limit theorems for functional “inverses” to study limit behavior of the empirical M-quantile process. We also show how to use these theorems to study asymptotics of regression quantiles.