Abstract

Multisample covariance estimation—that is, estimation of the covariance matrices associated with k distinct populations—is a classical problem in multivariate statistics. A common solution is to base estimation on the outcome of a test that these covariance matrices show some given pattern. Such a preliminary test may, for example, investigate whether or not the various covariance matrices are equal to each other (test of homogeneity), or whether or not they have common eigenvectors (test of common principal components), etc. Since it is usually unclear what the possible pattern might be, it is natural to consider a collection of such patterns, leading to a collection of preliminary tests, and to base estimation on the outcome of such a multiple testing rule. In the present work, we therefore study preliminary test estimation based on multiple tests. Since this is of interest also outside k-sample covariance estimation, we do so in a very general framework where it is only assumed that the sequence of models at hand is locally asymptotically normal. In this general setup, we define the proposed estimators and derive their asymptotic properties. We come back to k-sample covariance estimation to illustrate the asymptotic and finite-sample behaviors of our estimators. Finally, we treat a real data example that allows us to show their practical relevance in a supervised classification framework.

摘要

多样本协方差估计——即与k相关联的协方差矩阵的估计不同的总体——是多元统计中的经典问题。一个常见的解决方案是根据这些协方差矩阵显示某种给定模式的测试结果进行估计。例如，这样的初步测试可以调查各种协方差矩阵是否彼此相等（同质性测试），或者它们是否具有共同的特征向量（共同主成分测试）等。因为它通常是不清楚可能的模式是什么，很自然地考虑这些模式的集合，导致初步测试的集合，并根据这种多重测试规则的结果进行估计。因此，在目前的工作中，我们研究了基于多次测试的初步测试估计。因为这在k之外也很有趣-样本协方差估计，我们在一个非常通用的框架中这样做，其中仅假设手头的模型序列是局部渐近正态的。在这个一般设置中，我们定义了建议的估计量并推导出它们的渐近特性。我们回到k样本协方差估计来说明我们的估计量的渐近和有限样本行为。最后，我们处理一个真实的数据示例，使我们能够在监督分类框架中展示它们的实际相关性。