EFFICIENCY OF LINEAR ESTIMATORS UNDER HEAVY-TAILEDNESS: CONVOLUTIONS OF [alpha]-SYMMETRIC DISTRIBUTIONS
This paper focuses on the analysis of efficiency, peakedness, and majorization properties of linear estimators under heavy-tailedness assumptions. We demonstrate that peakedness and majorization properties of log-concavely distributed random samples continue to hold for convolutions of [alpha]-symmetric distributions with [alpha] > 1. However, these properties are reversed in the case of convolutions of [alpha]-symmetric distributions with [alpha] < 1.
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. In such a case, the sample mean exhibits monotone consistency, and an increase in the sample size always improves its performance. However, efficiency of the sample mean in the sense of peakedness decreases with the sample size if it is used to estimate the location parameter under extreme heavy-tailedness. We also present applications of the results in the study of concentration inequalities for linear estimators. a
c1 Address correspondence to Rustam Ibragimov, Department of Economics, Harvard University, Littauer Center, 1875 Cambridge St., Cambridge, MA 02138, USA; e-mail: email@example.com
a 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.” I am indebted to my advisers, 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, Aydin Cecen, Gary Chamberlain, Brian Dineen, Darrell Duffie, Xavier Gabaix, Tilmann Gneiting, Philip Haile, Wolfgang Härdle, Boyan Jovanovic, Samuel Karlin, Benoît Mandelbrot, Alex Maynard, Marcelo Morreira, Ingram Olkin, Ben Polak, Gustavo Soares, Kevin Song, and the participants at seminars at the Departments of Economics at Yale University, University of British Columbia, the University of California at San Diego, Harvard University, the London School of Economics and Political Science, Massachusetts Institute of Technology, the Université de Montréal, McGill University, and New York University, the Division of the Humanities and Social Sciences at California Institute of Technology, Nuffield College, University of Oxford, and the Department of Statistics at Columbia University, and the participants at the 18th New England Statistics Symposium at Harvard University, April 2004, the International Conference on Stochastic Finance, Lisbon, Portugal, September 2004, and the Conference “Heavy Tails and Stable Paretian Distributions in Finance and Macroeconomics” in celebration of the 80th birthday of Benoît Mandelbrot, Deutsche Bundesbank, Eltville, Germany, November 2005, for many helpful comments and discussions.