Nonparametric Density Estimation, Prediction, and Regression for Markov Sequences

Abstract

Let { X i } be a stationary Markov sequence having a transition probability density function f ( y | x ) giving the pdf of X i +1 | ( X i = x ). In this study, nonparametric density and regression techniques are employed to infer f ( y | x ) and m ( x ) = E [ X i + 1 | X i = x ]. It is seen that under certain regularity and Markovian assumptions, the asymptotic convergence rate of the nonparametric estimator m n ( x ) to the predictor m ( x ) is the same as it would have been had the X i 's been independently and identically distributed, and this rate is optimal in a certain sense. Consistency can be maintained after differentiability and even the Markovian assumptions are abandoned. Computational and modeling ramifications are explored. I claim that my methodology offers an interesting alternative to the popular ARMA approach.