Inferência em modelos de mistura de distribuições de transição

Abstract

In this dissertation, the problem of parameter estimation in Markov chains of order d will be addressed. In these chains the number of parameters to be estimated increases exponentially with d. When d is large, i.e. d is comparable to the sample size n, the estimation problem becomes much more complicated. In order to make estimation more feasible in these scenarios the model of Mixing Transition Distributions (MTD) was introduced by Raftery (1985). The MTD of order d is a parsimonious Markov model. Its fundamental hypothesis is that the conditional distribution of Xt given the past (Xt−1, . . . ,Xt−d) is expressed as a convex combination of distributions indexed by the relevant lags of the past. This way, the number of required parameters increases only linearly with d (instead of exponentially), becoming much smaller than in the general case for Markov chains of order d. MTD allows to determine individually which indices from the past have an influence on the present, and thus discard unnecessary information. In this work, consistent and computationally efficient methods to estimate the set of indices that influence the present will be discussed, especially in the high dimensional regime (i.e. when d is an increasing function of n). The performance of these methods will be verified through the quantification of the estimation error, and through simulated data, for different values of d and n.