Granularity and Efficiency
网格粒度与估计效率AI 解读 ↓
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网格粒度与估计效率
- 研究问题
- 当总体中心限定在均匀网格上时，舍入样本中位数与均值的渐近相对效率如何随分布形状、网格宽度和定义变化？
- 方法与思路
- 作者在均值与中位数一致的双侧扩展递增失效率分布类中，用大偏差理论比较舍入位置估计量的效率。
- 材料陈述的贡献
- 摘要称，效率对分布、网格和定义敏感；连续性结论因定义而异，且拉普拉斯情形的效率排序随网格粗细反转。
- 阅读关注点
- 应核对分布类和舍入规则、不同相对效率定义、大偏差率与有效方差的联系，以及连续性和排序反转的条件。
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- 仅据版本未注明的 OpenAlex 索引摘要；多处比较对象符号缺失，需查原PDF，不能重构公式或外推该分布类结论及刊本变化。
原文摘要
Hammersley (1950) considered, among other matters, the asymptotic relative efficiency (ARE) of the rounded sample median M ε with respect to the rounded sample mean as estimate of a Normal population mean restricted to a uniform grid of mesh size 2ε. This article extends Hammersley's work to a certain class of two-sided extended increasing failure rate (TEIFR) distributions for which the (grid-valued) population mean and median coincide, their common value designated as μ. The ARE of M ε with respect to , as estimators of μ, is examined for our class via the theory of large deviations. The role of the TEIFR assumption is simply to ensure that the tails of the distribution of X i − μ fall off quickly enough to make comparison of asymptotic probabilities of large (beyond ε) deviations of the location-normalized sample median M − μ and mean relevant to the comparison of their asymptotic variances. Even within our somewhat narrow class, we find the ARE of M ε with respect to surprisingly sensitive to distribution shape, as well as to grid mesh size and the actual definition of ARE. Among our findings is that, in the symmetric TEIFR class, the ARE of M ε with respect to is continuous in ε at ε = 0 under a definition of ARE closely related to the commonly used limiting ratio of equivalent sample sizes, but it is not continuous at ε = 0 under Hammersley's definition of ARE. A related finding is that, within the TEIFR class, the asymptotic effective variance [in the sense of Bahadur (1960)] of the sample median M equals its asymptotic variance as usually defined. Another finding is that, in the case of the Laplace distribution, M ε is asymptotically more efficient than , as estimator of the grid-valued population center μ, when the grid is fine (ε small), but it is asymptotically less efficient when the grid is coarse (ε large). All of these findings stem from the comparison of large-deviation rates that are equally relevant to the comparisons of asymptotic error rates of certain tests using and M as test statistics, a matter mentioned briefly in the last section.