Um estudo sobre a informação de fisher de grau q de uma variável aleatória discreta e suas relações com a informação logarítmica

Abstract

This work, guided by professor Franquiberto dos Santos Pessoa aims to complement requirements required by the Department of Mathematics of the Federal University of Ceará to grant the Master's degree in Pure Mathematics. This is a study in the discrete case of the Logarithmic Information of degree q linked to an extension of Fisher's Information that B. Bouchon and F. Pessoa [1] developed for the continuous case. It will be considered the probabilized space (omega, Q, P) where omega is a finite-dimensional Euclidean space, Q is the Borel-omega algebra, and P is the probability law over Q of a discrete random variable X given by Pi (theta), i belongs to I, where I is a finite or enumerable set of indices and theta an unknown real parameter. For the purpose of estimating a function g (theta) of the theta parameter we consider a statistic T and try to show the qualities of this estimator through Cramer-Rao type inequalities and necessarily lower limits that are Fisher's information functions in a much more context general than the classic Cramer-Rao formula. The conditions of regularity for the validity of the Cramer-Rao inequality will almost always be required, however, we will also obtain results without the requirement of such conditions.