Deviation moments of the substitution estimator and its piecewise smooth approximations (Q1963375)

Let \(X_1,\ldots,X_n\), \(X_i\in{\mathbb R}^d\), be a \(d\)-dimensional sample of not necessarily independent but identically distributed random vectors. Let \(t_n(x)=t_n(x,X_1,\ldots,X_n)\) be an \(s\)-dimensional vector statistic and let \(t(x)\) be a bounded \(s\)-dimensional vector-valued function. Given a function \(H\:{\mathbb R}^d\to{\mathbb R}\), the author considers the substitution statistic \(H(t_n(x))\) of the parameter \(H(t(x))\). Under certain conditions on \(H\), an estimate is presented for the moments \({\mathbb E}|H(t_n(x))-H(t(x))|^k\) in terms of the moments \({\mathbb E}|t_n(x)-t(x)|^k\). In addition, a number of estimates are presented for the difference \(|{\mathbb E}[H(t_n(x))-H(t(x))]^k-{\mathbb E}[\nabla H(t)(t_n(x)-t(x))]^k|\). Some applications are given to nonparametric estimation of the regression functions and to the estimation of ratios and products of two random variables.