$L$-Statistics in Complex Survey Problems

研究问题: 在分层多阶段调查抽样中，顺序统计量的线性组合及样本分位数如何进行渐近推断并一致估计其方差？
方法与思路: 作者区分平滑与非平滑 L 统计量，对前者建立渐近正态性和 Jackknife 方差估计，对分位数另行推导方差及区间性质。
材料陈述的贡献: 摘要称证明平滑 L 统计量及满足额外条件的分位数渐近正态，给出一致方差估计，并证明 Woodruff 分位数置信区间渐近有效。
阅读关注点: 应核对抽样设计与有限总体的渐近条件、平滑权重要求，以及分位数额外条件和 Jackknife、Woodruff 方法的具体实现。
阅读边界: 仅据版本未注明的 OpenAlex 索引摘要；弱条件与分位数所需额外条件未展开，也未报告有限样本覆盖，不应将所有分位数等同于平滑统计量处理。

GPT-6.1 Sol · 高 · 2026-10-05

原文摘要

We study linear combinations of order statistics ( $L$ -statistics) in survey problems with a stratified multistage sampling design. Two general types of $L$ -statistics are considered: smooth $L$ -statistics with weights generated by a smooth function and nonsmooth $L$ -statistics (sample quantiles). The trimmed sample mean, the decile mean and variance, the sample Lorenz curve and the sample Gini's family parameters are examples of smooth $L$ -statistics or functions of smooth $L$ -statistics used in survey problems. It is shown that under weak conditions the smooth $L$ -statistics are asymptotically normal and their asymptotic variances can be consistently estimated by jackknifing. For the sample quantiles, their asymptotic normality requires more conditions on the finite population distribution functions. Consistent estimators for the asymptotic variances of the sample quantiles are derived. Asymptotic validity of the Woodruff's confidence intervals for population quantiles is also proved.We study linear combinations of order statistics ($L$-statistics) in survey problems with a stratified multistage sampling design. Two general types of $L$-statistics are considered: smooth $L$-statistics with weights generated by a smooth function and nonsmooth $L$-statistics (sample quantiles). The trimmed sample mean, the decile mean and variance, the sample Lorenz curve and the sample Gini's family parameters are examples of smooth $L$-statistics or functions of smooth $L$-statistics used in survey problems. It is shown that under weak conditions the smooth $L$-statistics are asymptotically normal and their asymptotic variances can be consistently estimated by jackknifing. For the sample quantiles, their asymptotic normality requires more conditions on the finite population distribution functions. Consistent estimators for the asymptotic variances of the sample quantiles are derived. Asymptotic validity of the Woodruff's confidence intervals for population quantiles is also proved.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）

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分类依据与纠错

LLL-Statistics in Complex Survey Problems

代码、数据与补充材料

复杂调查抽样中的 L 统计量

原文摘要

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$L$ -Statistics in Complex Survey Problems