Variedades estatísticas generalizadas e uma modificação do estimador de máxima verossimilhança

Abstract

This thesis is devoted to study generalized statistical manifolds, within the context of geometry mainly and statistical manifold. Our principal contribution is the generalization of some results within these contexts cited before. Using the ϕ-function, which is used primarily in nonparametric families and has similarity with the exponential function, we parameterize the statistical manifold P, which is the focus of study here. From the ϕ- function, we define a function Dϕ(· k ·) between two probability distributions, called ϕ-divergence, with which we can define a metric gij and recover a pair of dual connections D(−1) e D(1) in P. With these connections, we define a family of connections D(α), which may also be recovered from a class of divergence Dϕ(α)(· k ·). Furthermore, we generalize the Rényi divergence and Kullback–Leibler divergence. In addition to the geometric aspects and considering that P is a parametric family, we will study questions about statistics inference. More precisely, we propose a more general interpretation of the method maximum likelihood estimator (MLE), which we define by ϕ(−1)-likelihood, in reference to MLE. This is developed for any ϕ-function given which we associate a likelihood function Lϕ. We will use an algorithm developed to solve the search problem of an equation roots that maximizes the likelihood function, in the presence of restriction not always find an analytical solution that depends of comple- xity of ϕ-function. Furthermore, some experimental results are showed. Finally, we study assumptions about asymptotic convergence estimators. Given these facts, the contribution of the thesis in geometric aspects is the generalization of exponential families, which will enable us to have more tools to study problems such as optimization, signal processing ans others. In aspects of statistical in- ference, we propose a more general version of estimation method where we can attack various problems.