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 a-symmetric distributions with a 1+ However, these properties are reversed in the case of convolutions of a-symmetric distributions with a 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+