Nonparametric Density Estimation, Prediction, and Regression for Markov Sequences

Abstract Let Xi be a stationary Markov sequence having a transition probability density function f (y | x) giving the pdf of X i +1 | (Xi = x). In this study, nonparametric density and regression techniques are employed to infer f (y | x) and m (x) = EX i + 1 | Xi = x. It is seen that under certain regularity and Markovian assumptions, the asymptotic convergence rate of the nonparametric estimator mn (x) to the predictor m (x) is the same as it would have been had the Xi '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.