ABSTRACT
Let X 1, …, X n be a random sample from a population with probability distribution P; the distribution is generally unknown, we only assume that its distribution function F belongs to some class https://www.w3.org/1998/Math/MathML"> F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315268125/d57fa0be-4eee-4d70-bfcd-ee2d3690ba91/content/inline-math3_1.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> of distribution functions. We look for an appropriate estimator of parameter θ, that can be expressed as a functional T(P) of P. The same parameter θ can be characterized by means of more functionals, e.g., the center of symmetry is simultaneously the expected value, the median, the modus of the distribution, and other possible characterizations. Some functionals T(P) are characterized implicitly as a root of an equation (or of a system of equations) or as a solution of a minimization (maximization) problem: such are the maximal likelihood estimator, moment estimator, etc. An estimator of parameter θ is obtained as an empirical functional, i.e., when one replaces P in the functional T(·) with the empirical distribution corresponding to the vector of observations X 1, …, X n .
