  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
$k\geq1$. 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
$k\geq2.5$. 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 $g$.
