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局部 U 统计过程与多个样本变量函数的密度估计AI 解读

A notion of local U-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for U-processes that may be useful in other contexts. This local U-statistic process is based on an estimator of the density of a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517–525] and, as a consequence, uniform in bandwidth central limit theorems in the sup…

核型函数估计量关于带宽的一致性AI 解读

We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya–Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.

用秩统计理论研究一般 Bootstrap 的一致性AI 解读

A general notion of a bootstrapped mean constructed by exchangeably weighting sample points is introduced. Consistency of this generalized bootstrapped mean, which includes proposals of Efron and Rubin among others, is proved by classical linear rank statistics theory. The consistency of generalized bootstrapped empirical and quantile processes is also established.

广义分位数过程AI 解读

For random vectors taking values in $\mathbb{R}^d$ we introduce a notion of multivariate quantiles defined in terms of a class of sets and study an associated process which we call the generalized quantile process. This process specializes to the well known univariate quantile process. We obtain functional central limit theorems for our generalized quantile process and show that both Gaussian and non-Gaussian limiting processes can arise. A number of interesting example are…

摘要暂缺

经验函数的自助法AI 解读

We develop the complete bootstrapped parallel to the asymptotic theory of weighted empirical and quantile processes. Utilizing this parallel theory, we present a general body of techniques to establish the asymptotic validity of the bootstrap method of constructing confidence bands for statistical functions. These techniques are demonstrated to be applicable to the construction of asymptotic bootstrap confidence bands for a variety of concrete functions.

基于 L 统计量平方和的位置—尺度分布族最优拟合优度检验AI 解读

A class of goodness-of-fit tests based on sums of squares of $L$-statistics is proposed for testing a composite parametric location and/or scale null hypothesis versus a general parametric alternative. It is shown that such tests can be constructed optimally to have the same asymptotic power against sequences of local alternatives as the generalized likelihood ratio statistics [G.L.R.S.] and, in fact, under suitable regularity conditions to be asymptotically equivalent to…

对尾部备择更敏感的修正 Kolmogorov–Smirnov 检验AI 解读

It is well known that the Kolmogorov-Smirnov (K-S) test exhibits poor sensitivity to deviations from the hypothesized distribution that occur in the tails. A modified version of the K-S test is introduced that is more sensitive than the K-S test to deviations in the tails. The finite and infinite sample distribution along with the consistency properties of the proposed test are studied. Tables of critical values are provided for two versions of the test (one sensitive to…

位置参数 L 估计权重函数的极小极大选择准则AI 解读

Let $X_1, \cdots, X_n$ be independent with common distribution $F$ symmetric about $\mu$. Let $T_n = n^{-1} \sum^n_{i = 1} J(i/(n + 1))X_{in}$ be an $L$-estimate of $\mu$ based on a weight function $J$ and the order statistics $X_{1n} \leq \cdots \leq X_{nn}$ of $X_1, \cdots, X_n$. Under very general regularity conditions $n^{1/2}T_n$ has asymptotic variance $\sigma^2(J, F)$. A weight function $J_0$ is found that minimizes the maximum of $\sigma^2(J, F)/s^2(F)$, whenever $s(F)$…

具有光滑评分函数的次序统计量线性组合的渐近正态性AI 解读

Asymptotic normality of linear combinations of order statistics of the form $T_n = n^{-1} \sum J(i/(n + 1))X_{in}$ is investigated along with a slightly trimmed version of $T_n$. Theorem 5 of Stigler (1974) is extended to show asymptotic normality of $T_n$ for a wide class of score functions. In addition, a proof of Theorem 4 of Stigler (1974) is given.

利用序贯秩统计量证明简单线性秩统计量的极限定理AI 解读

A technique is introduced to prove limit theorems for simple linear rank statistics by means of an approximating statistic based on sequential ranks. This approximation is shown to be close enough to prove asymptotic normality of simple linear rank statistics under the null hypothesis and to obtain a bound on the rate of convergence to normality when the score function is unbounded. In addition, a law of the iterated logarithm and an invariance principle are given for simple…