EFFICIENCY OF LINEAR ESTIMATORS UNDER HEAVY-TAILEDNESS: CONVOLUTIONS OF [alpha]-SYMMETRIC DISTRIBUTIONS
The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Ibragimov, Rustam. (2007). Efficiency of linear estimators under heavy-tailedness: convolutions of [alpha]-symmetric distributions. Econometric Theory 23(3): 501-517. Published Version ABSTRACT The present paper focuses on the analysis of efficiency, peakedness and majorization properties of linear estimators under heavy-tailedness assumptions. The main results
... ow that peakedness and majorization properties of log-concavely distributed random samples established by Proschan (1965) continue to hold for convolutions of 1 The results in this paper constitute a part of the author's dissertation "New majorization theory in economics and martingale convergence results in econometrics" presented to the faculty of the Graduate School of Yale University in candidacy for the degree of Doctor of Philosophy in Economics in March, 2005. Some of the results were originally contained in the work circulated in 2003-2005 under the titles "Shifting paradigms: On the robustness of economic models to heavy-tailedness assumptions" and "On the robustness of economic models to heavy-tailedness assumptions" 2 I am indebted to my advisors, Donald Andrews, Peter Phillips and Herbert Scarf, for all their support and guidance in all stages of the current project. I also thank the Associate Editor, two anonymous referees, Donald Brown, , for many helpful comments and discussions. α−symmetric distributions with α > 1. However, these properties are reversed in the case of convolutions of α−symmetric distributions with α < 1. Among other results, the paper shows 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. In addition, in such a case, the sample mean exhibits the property of monotone consistency and, thus, an increase in the sample size always improves its performance. However, 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 heavy-tailedness. The paper also provides applications of the main efficiency and majorization comparison results in the study of concentration inequalities for linear estimators.