Asymptotic Linearity of Wilcoxon Signed-Rank Statistics
Wilcoxon符号秩统计量的渐近线性性AI 解读 ↓
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Wilcoxon符号秩统计量的渐近线性性
- 研究问题
- 在单参数一般线性回归模型中，Wilcoxon型符号秩统计量能否获得更强的渐近线性描述，以刻画估计量的极限行为？
- 方法与思路
- 作者研究统计量与参数线性形式之差的标准化弱收敛，使其收敛到线性过程，并讨论相关估计应用。
- 材料陈述的贡献
- 摘要称，论文建立比已有渐近线性表述更强的过程结果，可用于密度L2范数平方估计及线性化版本的渐近分析。
- 阅读关注点
- 应核对线性过程所处的函数空间、标准化方式与正则条件，并区分统计量线性化结果和估计量分布结论。
- 阅读边界
- 仅据OpenAlex索引摘要，稿件版本未注明；摘要未列出完整假设、极限过程形式或效率结果，不能据此判断稳健性范围。
原文摘要
As a "robust" alternative to the least squares estimates for a regression parameter, Koul (1969) proposed new estimates based on signed-rank statistics. To find out their asymptotic distribution Koul proved that under quite general assumptions, the signed-rank statistics of Wilcoxon type are asymptotically linear in the sense that they are uniformly approximable by linear forms in the regression parameter. More general results have been obtained by Van Eeden (1972) in a paper which is an analog to Jureckova's paper (1969) dealing with linear rank statistics. In 1972 the author proved that the statistic used to define the Hodges-Lehmann estimate for a location parameter is asymptotically linear in a stronger sense, the result being to Koul's theorem what the central limit theorem is to the weak law of large numbers. For the general linear regression model with one parameter the signed-rank statistics are proved to be linear in a strong sense, that is, the differences between the statistics and the linear forms mentioned above, properly normalized, converge weakly to linear processes. Results in this direction for linear rank statistics have been obtained by Jureckova (1973). As an application of the theorems presented here, one can construct new estimates for the squared $L_2$-norm of the underlying density, and this in much the same way as in Antille (1974). It is also possible to get more information about the asymptotic behavior of the linearized versions, proposed by Kraft and Van Eeden (1972).