On -characteristic functions and applications to asymptotic statistical inference

ABSTRACT

In this paper, we give emphasis to a method to do statistical inference and to study properties of random variables, whose probability density functions (pdfs) do not possess good regularity, decay, and integrability properties. The main tool will be what we will call -characteristic function, a generalization of the classical characteristic function that is basically a measurable transform of pdfs. In this perspective and using this terminology, we will restate and prove theorems, such as the law of large numbers and the central limit theorem that now, after this measurable transform, apply to basically every distribution, upon the correct choice of a free parameter α. We apply this theory to hypothesis testing and to the construction of confidence intervals for location parameters. We connect the classical parameters of a distribution to their related 1/α-counterparts, that we will call -momenta. We treat in detail the case of the multivariate Cauchy distribution for which we compute explicitly all the -expected values and -variances in dimension n = 1 and for which we construct an approximate confidence interval for the location parameter μ, by means of asymptotic theorems in the -context. Among the other things and to illustrate the usefulness of this point of view, we prove some new characterizations of the Poisson distribution, the uniform discrete, and the uniform continuous distribution.

KEYWORDS:

• Asymptotic inference
• central limit theorem
• characteristic functions
• characterization of distributions
• hypothesis testing
• weak law of large numbers

MATHEMATICS SUBJECT CLASSIFICATION:

• 60E10
• 60E05
• 60F05
• 62E20
• 62H15
• 60E15

Acknowledgments

I want to thank my family and Victoria Ban for their constant support. I thank also my supervisor Prof. Narayanaswamy Balakrishnan for his constant help and inspiring guidance. I warmly thank the referees of this paper, whose comments have led to several improvements in both the content and exposition.