Weak convergence of the maximum error of the bootstrap quantile estimate
Let \(F_ n\) be the empirical d.f. pertaining to a sample \(X_ 1,...,X_ n\) of independent random variables with common d.f. F and \(F^*_ n\) the empirical d.f. pertaining to a sample \(X^*_ 1,...,X^*_ n\) of independent random variables with common d.f. \(F_ n\). Furthermore, let \(G^{-1}(q)\) be the q-quantile of a d.f. G. Then, the bootstrap estimate of \(P\{F_ n^{-1}(q)-F^{-1}(q)\leq t\}\) is given by \(P_ n\{F_ n^{*-1}(q)-F_ n^{-1}(q)\leq t\}\). Define the stochastic process \((Z_ n(t))_{t\in R}\) with values in \(D_ R\) by \[ Z_ n(t)=P_ n\{n^{1/2}(F_ n^{*-1}(q)-F_ n^{-1}(q))\leq t\}- P\{n^{1/2}(F_ n^{-1}(q)-F^{-1}(q))\leq t\} \] where \(D_ R\) denotes the space of real valued functions on R which are right continuous and have left limits. The author obtains a weak convergence theorem of the random variable \(n^{1/4}\sup_{t\in R} | Z_ n(t)|\) under appropriate conditions on F. The result does not follow from the result on weak convergence in \(D_ R\) of the stochastic process \((n^{1/4}Z_ n(t))_{t\in R}\) which was obtained by the author and \textit{R.-D. Reiss} [Ann. Probab. 17, No.1, 362-371 (1989; Zbl 0684.62036)].
scientific article; zbMATH DE number 4169855