On Efficiencey of Linear Estimators Under Heavy-Tailedness

Abstract

The present paper develops a new unified approach to the analysis of efficiency, peakedness and majorization properties of linear estimators. It further studies the robustness of these properties to heavy-tailedness assumptions. the main results show that peakedness and majorization phenomena for random samples from log-concavely distributed populations established in the seminal work by Proschan (1965) continue to hold for not extremely thick- tailed distributions. However, these phenomena are reversed in the case of populations with extremely heavy-tailed densities. Among other results, we show that the sample mean is the best linear unbiased estimator of the population mean for not extremely heavy-tailed populations in the sense of its peakedness properties. Moreover, in such a case, the sample mean exhibits the important property of monotone consistency and, thus, an increase in the sample size always improves its performance. However, as we demonstrate, efficiency of the sample mean in the sense of its peakedness decreases with the sample size if the sample mean is used to estimate the population center under extreme thick-tailedness. We also provide applications of the main efficiency and majorization comparison results in the study of concentration inequalities for linear estimators as well as their extensions to the case of wide classes of dependent data. The main results obtained in the paper provide the basis for the analysis of many problems in a number of other areas, in addition to econometrics and statistics, and, in particular, have applications in the study of robustness of model of firm growth for firms that can invest into information about their markets, value at risk analysis, optimal strategies for a multiproduct monopolist as well that of inheritance models in mathematical evolutionary theory.