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Grenander型估计量上确界范数误差的极限分布
Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n/\log n)^{-1/3}$ and the limiting distribution to be Gumbel.
递减密度的非参数最大似然估计在支撑边界附近的行为
We investigate the behavior of the nonparametric maximum likelihood estimator f̂n for a decreasing density f near the boundaries of the support of f. We establish the limiting distribution of f̂n(n−α), where we need to distinguish between different values of 0<α<1. Similar results are obtained for the upper endpoint of the support, in the case it is finite. This yields consistent estimators for the values of f at the boundaries of the support. The limit distribution of these…
Grenander 估计量的 Lk 误差渐近正态性
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…