Quantile estimation and the statistical relative efficiency curve

摘要

In this article, we introduce a new practical tool-relative efficiency curve (REC)-for comparison of two competing statistical procedures. While in other scientific areas the term of REC has been around for some time, in statistics it seems to be new. In estimation, the curve is constructed by employing asymptotic properties of quantile estimators. Suppose two consistent and asymptotically normal estimators of a fixed quantile of the underlying distribution are available. Plotting of the ratio of their variances versus quantiles at various probability levels yields an REC. Such a curve provides information about the accuracy of one estimator relative to another when both are designed to estimate the same (fixed but arbitrary) quantile of the distribution. Thus, depending on the objective of application, the REC can help one choose between parametric, robust parametric, empirical nonparametric or other method of estimation for the measure of interest. Further, other possibilities for defining (statistical) RECs are also discussed, and illustrative examples for (equivalent) Pareto and exponential, and lognormal and normal distributions are provided. Specifically, graphs of RECs of maximum likelihood, method of trimmed moments, and empirical nonparametric estimators of distribution quantiles are presented.