Finite sample nonparametric inference and large sample efficiency

Finite sample nonparametric inference and large sample efficiency
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有限样本非参数推理和大样本效率

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发表时间：
1998
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通讯作者：
Joseph P. Romano
Joseph P. Romano
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作者：
Joseph P. Romano

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给定来自分布$F$的样本$X_1，\DETS，X_n$， 关于均值的非参数可信区间的构造问题 $\MU(F)$被考虑。与引导过程或基于正常的引导过程不同 中的任何过程都是真正的非参数过程。 可信度区间包含以$\MU(F)$为基础的概率 对于来自$F$的大小为$n$的样本，对于所有$F$和所有 $n$。Bahadur和Savage证明了不可能找到有效的(或 在没有某些限制的情况下，$\MU(F)$的有界)可信区间。因此，我们 假设$F$在已知紧凑集合上受支持，我们将其视为$[0， 1]$。在这种设置下，获得了一个渐近效率结果，该结果给出了一个 任何保守区间大小的下界。然后，我们提供一个 构造在水平上满足我们的有限样本要求的区间， 但仍具有渐近效率性质。因此，使用的代价是 完全非参数程序在考虑Exact和Exact 推理语句和渐近效率可以忽略不计。很大一部分是 对于平均值的完成也适用于其他设置。
Given a sample $X_1,\dots, X_n$ from a distribution $F$, the problem of constructing nonparametric confidence intervals for the mean $\mu(F)$ is considered. Unlike bootstrap procedures or those based on normal approximations, we insist on any procedure being truly nonparametric in the sense that the probability that the confidence interval contains $\mu(F)$ based on a sample of size $n$ from $F$ be at least $1 - \alpha$ for all $F$ and all $n$. Bahadur and Savage proved it is impossible to find an effective (or bounded) confidence interval for $\mu(F)$ without some restrictions. Thus,we assume that $F$ is supported on a known compact set, which we take to be $[0, 1]$. In this setting, an asymptotic efficiency result is obtained that gives a lower bound on the size of any conservative interval. We then provide a construction of an interval that meets our finite sample requirement on level, yet has an asymptotic efficiency property. Thus, the price to be paid for using fully nonparametric procedures when considering the trade-off between exact inference statements and asymptotic efficiency is negligible. Much of what is accomplished for the mean generalizes to other settings as well.