Estimates of Location Based on Rank Tests

研究问题: 经典线性模型或正态假设下的位置估计对粗差敏感，而秩检验在检验问题中已展现稳健性。本文的核心问题是：能否直接由秩检验统计量构造位置或平移参数的点估计，并使其具备稳健性、无偏性及可计算的渐近分布？
方法与思路: 将 Wilcoxon 或正态得分等秩检验统计量反演为点估计：对每个候选参数值，计算相应秩检验统计量，取使检验统计量等于某基准值（如零）的参数值作为估计。该方法把检验的稳健性传递到估计，避免直接删除或缩尾异常值。
摘要陈述的贡献: 摘要明确：在 2–5 节定义基于秩检验统计量的位置或平移参数估计；6–7 节证明正则性与不变性；8 节证明对称分布下估计分布关于参数对称（因而无偏）；9 节给出非对称下小样本中位无偏及与置信水平 1/2 置信限的密切关系；10 节给出大样本渐近正态性；11 节证明估计对经典线性估计的渐近相对效率等于对应秩检验对 t 检验的 Pitman 效率。
预备知识: 需先掌握：秩检验基本思想（Wilcoxon 符号秩、两样本 Wilcoxon、正态得分）、位置与平移参数、估计的无偏性与中位无偏性、渐近正态性与渐近相对效率、Pitman 效率、置信区间与假设检验的对偶关系。对分布对称性、不变性等概念也应有了解。
阅读重点: 重点核对：估计定义中检验统计量反演的具体形式与基准值选择；正则性与不变性证明所依赖的假设；对称性假设在无偏性证明中的作用；非对称下中位无偏是精确还是近似、小样本条件；渐近正态性所需的正则条件；与 1/2 置信限重合的条件；Pitman 效率等价结论的适用范围。
阅读边界: 基于原文摘要，未阅读全文。摘要未给出具体数据、数值结果或完整识别条件；上述方法细节与结论均以摘要陈述为限，阅读原文时应核对各节假设与推导。

DeepSeek Flash · 2026-09-30 · 摘要解读

原文摘要

A serious objection to many of the classical statistical methods based on linear models or normality assumptions is their vulnerability to gross errors. For certain testing problems this difficulty is successfully overcome by rank tests such as the two Wilcoxon tests or the Kruskal-Wallis H-test. Their power is more robust against gross errors than that of the $t$- and $F$-tests, and their efficiency loss is quite small even in the rare case in which the suspicion of the possibility of gross errors is unfounded. For the corresponding problems of point estimation a beginning has been made to attack the difficulty by modifying the classical estimates either through removal or Winsorization of outlying observations; see for example Tukey (1960) and Anscombe (1960). It is the purpose of the present paper to explore a different approach to these problems of point estimation. In Sections 2-5 point estimates of location or shift parameter are defined in terms of rank test statistics such as the Wilcoxon or normal scores statistic, which are successful in providing robust power for the corresponding testing problems. In Sections 6 and 7, certain regularity and invariance properties of these estimates are proved. The distributions of the estimates are shown in Section 8 to be symmetric with respect to the parameter being estimated--and hence in particular to be unbiased--if the underlying distribution of the observations on which the estimate is based is symmetric. Without this assumption, the estimates are shown in Section 9 to be either exactly or approximately median unbiased for small samples and in Section 10 to be approximately normally distributed about the true parameter value for large samples. The variance of this asymptotic distribution depends of course on the underlying distribution of the observations, so that the estimates are not "distribution-free." In Section 9 there is also established a close relationship between the estimates and the corresponding upper and lower confidence bound for the parameter at confidence level $\frac{1}{2}$, with which the estimate coincide in many cases. Finally, in Section 11, it is proved that the asymptotic relative efficiency of the estimates to the classical linear estimates is the same as the Pitman efficiency of the rank tests on which they are based to the corresponding $t$-tests.

摘要来源：OpenAlex

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基于秩检验的位置参数估计

研究问题

经典线性模型或正态假设下的位置估计对粗差敏感，而秩检验在检验问题中已展现稳健性。本文的核心问题是：能否直接由秩检验统计量构造位置或平移参数的点估计，并使其具备稳健性、无偏性及可计算的渐近分布？

方法与思路

将 Wilcoxon 或正态得分等秩检验统计量反演为点估计：对每个候选参数值，计算相应秩检验统计量，取使检验统计量等于某基准值（如零）的参数值作为估计。该方法把检验的稳健性传递到估计，避免直接删除或缩尾异常值。

摘要陈述的贡献

摘要明确：在 2–5 节定义基于秩检验统计量的位置或平移参数估计；6–7 节证明正则性与不变性；8 节证明对称分布下估计分布关于参数对称（因而无偏）；9 节给出非对称下小样本中位无偏及与置信水平 1/2 置信限的密切关系；10 节给出大样本渐近正态性；11 节证明估计对经典线性估计的渐近相对效率等于对应秩检验对 t 检验的 Pitman 效率。

预备知识

需先掌握：秩检验基本思想（Wilcoxon 符号秩、两样本 Wilcoxon、正态得分）、位置与平移参数、估计的无偏性与中位无偏性、渐近正态性与渐近相对效率、Pitman 效率、置信区间与假设检验的对偶关系。对分布对称性、不变性等概念也应有了解。

阅读重点

重点核对：估计定义中检验统计量反演的具体形式与基准值选择；正则性与不变性证明所依赖的假设；对称性假设在无偏性证明中的作用；非对称下中位无偏是精确还是近似、小样本条件；渐近正态性所需的正则条件；与 1/2 置信限重合的条件；Pitman 效率等价结论的适用范围。

阅读边界

基于原文摘要，未阅读全文。摘要未给出具体数据、数值结果或完整识别条件；上述方法细节与结论均以摘要陈述为限，阅读原文时应核对各节假设与推导。

DeepSeek Flash · 2026-09-30 · 摘要解读

原文摘要

t

- and

F

-tests, and their efficiency loss is quite small even in the rare case in which the suspicion of the possibility of gross errors is unfounded. For the corresponding problems of point estimation a beginning has been made to attack the difficulty by modifying the classical estimates either through removal or Winsorization of outlying observations; see for example Tukey (1960) and Anscombe (1960). It is the purpose of the present paper to explore a different approach to these problems of point estimation. In Sections 2-5 point estimates of location or shift parameter are defined in terms of rank test statistics such as the Wilcoxon or normal scores statistic, which are successful in providing robust power for the corresponding testing problems. In Sections 6 and 7, certain regularity and invariance properties of these estimates are proved. The distributions of the estimates are shown in Section 8 to be symmetric with respect to the parameter being estimated--and hence in particular to be unbiased--if the underlying distribution of the observations on which the estimate is based is symmetric. Without this assumption, the estimates are shown in Section 9 to be either exactly or approximately median unbiased for small samples and in Section 10 to be approximately normally distributed about the true parameter value for large samples. The variance of this asymptotic distribution depends of course on the underlying distribution of the observations, so that the estimates are not "distribution-free." In Section 9 there is also established a close relationship between the estimates and the corresponding upper and lower confidence bound for the parameter at confidence level

\frac{1}{2}

, with which the estimate coincide in many cases. Finally, in Section 11, it is proved that the asymptotic relative efficiency of the estimates to the classical linear estimates is the same as the Pitman efficiency of the rank tests on which they are based to the corresponding

t

-tests.A serious objection to many of the classical statistical methods based on linear models or normality assumptions is their vulnerability to gross errors. For certain testing problems this difficulty is successfully overcome by rank tests such as the two Wilcoxon tests or the Kruskal-Wallis H-test. Their power is more robust against gross errors than that of the $t$- and $F$-tests, and their efficiency loss is quite small even in the rare case in which the suspicion of the possibility of gross errors is unfounded. For the corresponding problems of point estimation a beginning has been made to attack the difficulty by modifying the classical estimates either through removal or Winsorization of outlying observations; see for example Tukey (1960) and Anscombe (1960). It is the purpose of the present paper to explore a different approach to these problems of point estimation. In Sections 2-5 point estimates of location or shift parameter are defined in terms of rank test statistics such as the Wilcoxon or normal scores statistic, which are successful in providing robust power for the corresponding testing problems. In Sections 6 and 7, certain regularity and invariance properties of these estimates are proved. The distributions of the estimates are shown in Section 8 to be symmetric with respect to the parameter being estimated--and hence in particular to be unbiased--if the underlying distribution of the observations on which the estimate is based is symmetric. Without this assumption, the estimates are shown in Section 9 to be either exactly or approximately median unbiased for small samples and in Section 10 to be approximately normally distributed about the true parameter value for large samples. The variance of this asymptotic distribution depends of course on the underlying distribution of the observations, so that the estimates are not "distribution-free." In Section 9 there is also established a close relationship between the estimates and the corresponding upper and lower confidence bound for the parameter at confidence level $\frac{1}{2}$, with which the estimate coincide in many cases. Finally, in Section 11, it is proved that the asymptotic relative efficiency of the estimates to the classical linear estimates is the same as the Pitman efficiency of the rank tests on which they are based to the corresponding $t$-tests.

摘要来源：OpenAlex