Asymptotic normality of the Lk-error of the Grenander estimator

原文摘要

We investigate the limit behavior of the \(L_k\)-distance between a decreasing density \(f\) and its nonparametric maximum likelihood estimator \(\hat{f}_n\) for \(k \geq 1\). Due to the inconsistency of \(\hat{f}_n\) at zero, the case \(k = 2.5\) turns out to be a kind of transition point. We extend asymptotic normality of the \(L_1\)-distance to the \(L_k\)-distance for \(1 \leq k < 2.5\), and obtain the analogous limiting result for a modification of the \(L_k\)-distance for \(k \geq 2.5\). Since the \(L_1\)-distance is the area between \(f\) and \(\hat{f}_n\), which is also the area between the inverse \(g\) of \(f\) and the more tractable inverse \(U_n\) of \(\hat{f}_n\), the problem can be reduced immediately to deriving asymptotic normality of the \(L_1\)-distance between \(U_n\) and \(g\). Although we lose this easy correspondence for \(k > 1\), we show that the \(L_k\)-distance between \(f\) and \(\hat{f}_n\) is asymptotically equivalent to the \(L_k\)-distance between \(U_n\) and \(g\).

摘要来源：TU Eindhoven 大学仓储期刊版 PDF 摘要