On the rate of convergence of empirical distribution functions in \(AR(1)\) models
The author considers an \(AR(1)\) process \(U_k=\beta U_{k-1}+\varepsilon_k\), where \(\varepsilon_k\) are i.i.d. with d.f. \(G\). The unknown \(G\) is estimated by the empirical distribution function of the residuals \(G_n\). The author derives an upper bound for NEWLINE\[NEWLINE\Pr\{\sup_x\sqrt{n}|G_n(x)-G(x)|>\varepsilon\},NEWLINE\]NEWLINE for the case when NEWLINE\[NEWLINE\sup_x|G''(x)|<\infty,\quad E\varepsilon_n^4<\infty,\quad |\beta|\leq\Lambda<1.NEWLINE\]
scientific article; zbMATH DE number 1646310