ABSTRACT
This chapter presents characteristic function (CF) based tools as an approach for investigating the properties of distribution functions (DF) and their associated quantities. The form of a CF can easily define a family of DFs, thus generalizing known conjugate families. This is very important in parametric statistics and the construction of Bayesian priors . A CF can mathematically represent a random variable in terms of its given properties even when the DF does not have an explicit analytical form. Oftentimes we need to estimate DFs based on observations subject to noise effects or that are based on sums or maximums. These scenarios and various tasks of sequential procedures, as well as in the context of renewal theory, are examples of scenarios where CFs and Laplace transformations can play a main role in analytical analyses. Chapter 2 considers: the properties of CFs; convolution problems; the one-to-one mapping propositions; Tauberian Theorems ; risk-efficient estimation; the law of large numbers; the central limit theorem; issues of reconstructing the general distribution based on the distribution of some statistics; extensions and estimations of families of DFs; measurement error problems; cost-efficient designs; principles of Monte Carlo simulation and corresponding R codes.
