On 1α$\frac{1}{\alpha }$-characteristic functions and applications to asymptotic statistical inference

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 1α$\frac{1}{\alpha }$-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 1α$\frac{1}{\alpha }$-momenta. We treat in detail the case of the multivariate Cauchy distribution for which we compute explicitly all the 1α$\frac{1}{\alpha }$-expected values and 1α$\frac{1}{\alpha }$-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 1α$\frac{1}{\alpha }$-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.