Teoria dos grupos na estimação equivariante

Abstract

One concern that arises in scientific research is to gain information about the population under study, which can be made using a probability density function (pdf) that describes it. In turn, a pdf is characterized by unknown numerical quantities, called parameters. Thus, knowledge of population parameters generates information about the population that resides in a particular interest. There are different methods for finding estimators and often these methods provide different estimators for the same population parameter. Thus arises the need for methods to evaluate these estimators and, as the amount of possible estimators can be very large, a procedure to be adopted is to limit the class of estimators based on some propertie. In that sense, the principle of equivariance can be used for this purpose. The principle of equivariance is a principle that preserves some important features of the model and is widely used in classical statistics and closely related to important Group Theory in mathematics. In this context, the general objectives of this work consists of: i) present the definitions and theoretical results necessary for the development of the Equivariant Estimation Theory, ii) exemplify the use of equivariant estimators with an application of maximum rainfall data from Piracicaba-SP. In theoretical development, we determined the Pitman estimators for location and scale parameters, which are the estimators with even smaller risk in the class of equivariant estimators. The simultaneous estimation of scale and location parameters on location-scale families was also discussed. It has been shown that the Pitman estimators for location and scale parameters can be obtained as generalized Bayes estimators using appropriate priors. The admissibility and minimax properties of Pitman estimators were discussed. In the last section we have performed an application of equivariant estimators in the study of maximum rainfall in Piracicaba-SP.