Admissibility in Discrete and Continuous Invariant Nonparametric Estimation Problems and in their Multinomial Analogs
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离散、连续不变非参数估计及多项分布对应问题的可容许性
- 研究问题
- 经验分布函数、中位数及最优不变估计量，在连续、离散和多项分布对应问题中，何时具有可容许性？
- 方法与思路
- 作者构造离散和多项分布对应问题，考察不同分布函数损失与中位数不变损失，并研究已知端点下的估计量修正。
- 材料陈述的贡献
- 摘要称可容许性依损失与样本空间而变：离散经验分布函数在指定加权损失下可容许，连续偶数样本中位数则不然。
- 阅读关注点
- 核对各项结论对应的损失、参数空间与样本量，特别区分最优不变性、端点修正和可容许性的不同作用。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；部分离散情形的边界未列全，可容许性也不等于有限样本风险普遍最低。
原文摘要
Discrete and multinomial analogs are defined for classical (continuous) invariant nonparametric problems of estimating the sample cumulative distribution function (sample c.d.f.) and the sample median. Admissibility of classical estimators and their analogs is investigated. In discrete (including multinomial) settings the sample c.d.f. is shown to be an admissible estimator of the population c.d.f. under the invariant weighted Cramer-von Mises loss function $L_1(F, \hat{F}) = \int \big\lbrack (F(t) - \hat{F}(t))^2/(F(t)(1 - F(t))) \big\rbrack dF(t).$ Ordinary Cramer-von Mises loss--$L_2(F, \hat{F}) = \int \lbrack (F(t) - \hat{F}(t))^2 \rbrack dF(t)$--is also studied. Admissibility of the best invariant estimator is investigated. (It is well known in the classical problem that the sample c.d.f. is not the best invariant estimator, and hence is not admissible.) In most discrete settings this estimator must be modified in an obvious fashion to take into account the end points of the known domain of definition for the sample c.d.f. When this is done the resulting estimator is shown to be admissible in some of the discrete settings. However, in the classical continuous setting and in other discrete settings, the best invariant estimator, or its modification, is shown to be inadmissible. Kolmogorov-Smirnov loss for estimating the population c.d.f. is also investigated, but definitive admissibility results are obtained only for discrete problems with sample size 1. In discrete settings the sample median is an admissible estimator of the population median under invariant loss. In the continuous setting this is not true for even sample sizes.