Compromise Maximum Likelihood Estimators for Location
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位置参数的折中最大似然估计量
- 研究问题
- 如何构造位置参数估计量，使其在小样本中兼顾多种可能的密度形状，并在高斯数据、厚尾分布和异常值情形间取得稳健折中？
- 方法与思路
- 作者按样本似然对多个候选密度的最大似然估计进行加权，形成 CMLE，并以高斯与 slash 分布构造的估计量开展模拟。
- 材料陈述的贡献
- 摘要报告，在所考察的小样本和分布设置中，高斯与 slash 折中估计优于比较的 M 估计及截尾均值，并展示股票组合收益实例。
- 阅读关注点
- 应核对似然权重的定义、候选密度的选择依据、比较性能的指标，以及样本量和分布范围对模拟结论的限制。
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原文摘要
This article describes a new approach for constructing robust location estimators. These estimators are constructed to be nearly optimal in small sample sizes simultaneously for two or more possible underlying shapes of the density of the data. The estimators are weighted averages of the maximum likelihood estimators for the densities considered, where the weights depend on the sample through the likelihood functions. Because of this relationship to the usual MLE's, these estimators are called compromise maximum likelihood estimators (CMLE's). For the CMLE's to exhibit good robustness properties, the densities used to construct the CMLE's should be chosen to, in a sense, “span” a reasonable range of possible underlying distributions of the data encountered in practice. For example, to construct CMLE's with the usual robustness properties, that is, which perform well both for narrow-tailed Gaussian data and for data containing outliers or from wide-tailed distributions, the CMLE's might be constructed based on the Gaussian distribution and a wide-tailed distribution such as the slash distribution. One would then expect that the CMLE's would perform well for data from all symmetric distributions with tails that are in between these two extremes. To verify that this is the case, Monte Carlo results are presented for CMLE's that compromise between the Gaussian and slash distributions for sample sizes 5, 10, and 20. Results are presented both for the Gaussian and slash distributions, which are used to construct the estimators, and for Student's t distributions with 2, 3, 5, 7, and 10 df, which have tails in between the Gaussian and slash extremes. The results show that the Gaussian-slash CMLE's perform better than the usual robust location estimators such as the M estimators and trimmed means. As an example of application of the estimators to real data, the behavior of the estimators for some stock portfolio return data is also examined.