1 research outputs found
Properties of complex-valued power means of random variables and their applications
We consider power means of independent and identically distributed (i.i.d.)
non-integrable random variables. The power mean is an example of a homogeneous
quasi-arithmetic mean. Under certain conditions, several limit theorems hold
for the power mean, similar to the case of the arithmetic mean of i.i.d.
integrable random variables. Our feature is that the generators of the power
means are allowed to be complex-valued, which enables us to consider the power
mean of random variables supported on the whole set of real numbers. We
establish integrabilities of the power mean of i.i.d. non-integrable random
variables and a limit theorem for the variances of the power mean. We also
consider the behavior of the power mean as the parameter of the power varies.
The complex-valued power means are unbiased, strongly-consistent, robust
estimators for the joint of the location and scale parameters of the Cauchy
distribution.Comment: 43 pages; v3: Section 2 for backgrounds and Section 8 for the mixture
Cauchy model added. Introduction shortened. To appear in Acta Math. Hun