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.