Asymptotic normality of the $L_k$-error of the Grenander estimator

We investigate the limit behavior of the $L_k$-distance between a decreasing density $f$ and its nonparametric maximum likelihood estimator ${\widehat{f}}_n$ for $k\ge 1$. Due to the inconsistency of ${\widehat{f}}_n$ 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 $1\le k<2.5$, and obtain the analogous limiting result for a modification of the $L_k$-distance for $k\ge 2.5$. Since the $L_1$-distance is the area between $f$ and ${\widehat{f}}_n$, which is also the area between the inverse $g$ of $f$ and the more tractable inverse $U_n$ of ${\widehat{f}}_n$, 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 ${\widehat{f}}_n$ is asymptotically equivalent to the $L_k$-distance between $U_n$ and $g$.