Ratio‐consistency of some invariant U ‐statistic‐based estimators with an application to high‐dimensional data ranking

ABSTRACT Ratio‐consistency is a very desirable property of estimators, especially in high‐dimensional statistics. In this article, unbiased and rotation‐translation‐invariant estimators based on linear combinations of U ‐statistics for some functions of the covariance matrix are derived using a general procedure. A useful result for showing the ratio‐consistency of such a kind of estimators utilizing the formula for the variance of U ‐statistics is provided, and sufficient conditions for the ratio‐consistency of these proposed unbiased estimators are then deduced. As an application, a novel procedure using these invariant estimators to rank high‐dimensional data is proposed.