Abstract :
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+
