Asymptotic normality of the Lk-error of the Grenander estimator
Asymptotic normality of the Lk-error of the Grenander estimator
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DOI：
10.1214/009053605000000462
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发表时间：
2006-02
影响因子：
10.3
通讯作者：
V. Kulikov;H. P. Lopuhaa
中科院分区：
文献类型：
--
作者：
V. Kulikov;H. P. Lopuhaa
本文研究了递减密度f与其非参数极大似然估计量n之间L_k-距离的极限行为。由于$\hat{f}_n$在零处的不一致性，k=2.5$的情况被证明是一种过渡点。本文将$L_1$-距离的渐近正态性推广到$L_k$-距离，证明了$f$和$\hat{f}_n$之间的$L_k$-距离渐近等价于$U_n$和$g$之间的$L_k$-距离.
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 $1\leq k 1$, 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 $g$.