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单阶段抽样设计的函数型中心极限定理AI 解读

For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz–Thompson empirical process and the Hájek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard…

Grenander型估计量上确界范数误差的极限分布AI 解读

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AI 解读

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AI 解读

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…

多元位置与散布再加权估计量的渐近性质AI 解读

We investigate the asymptotic behavior of a weighted sample mean and covariance,where the weights are determined by the Mahalanobis distances with respect to initial robust estimators.We derive an explicit expansion for the weighted estimators. From this expansion it can be seen that reweighting does not improve the rate of convergence of the initial estimators.We also show that if one uses smooth $S$-estimators to determine the weights, the weighted estimators are…

Groeneboom introduced a jump process that can be used (among other things) to study the asymptotic properties of the Grenander estimator of a monotone density. In this paper we derive the asymptotic normality of a suitably rescaled version of the $L_1$ error of the Grenander estimator, using properties of this jump process.