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) = E[X 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.

[1]  J. Doob.  Stochastic processes , 1953 .

[2]  G. S. Watson,et al.  Smooth regression analysis , 1964 .

[3]  G. Roussas.  Nonparametric Estimation of the Transition Distribution Function of a Markov Process , 1969 .

[4]  Eugene F. Schuster,et al.  Joint Asymptotic Distribution of the Estimated Regression Function at a Finite Number of Distinct Points , 1972 .

[5]  Sidney Yakowitz,et al.  A stochastic model for daily river flows in an arid region , 1973 .

[6]  Chester C. Kisiel,et al.  Procedures for determining the order of dependence in streamflow records , 1974 .

[7]  Sidney Yakowitz,et al.  Small-Sample Hypothesis Tests of Markov Order, with Application to Simulated and Hydrologic Chains , 1976 .

[8]  Robert E. Kalaba,et al.  Computational Probability and Simulation. , 1978 .

[9]  J. L. Denny,et al.  Admissible Run-Contingency Type Tests for Independence and Markov Dependence , 1978 .

[10]  Sidney Yakowitz,et al.  A nonparametric Markov Model for daily river flow , 1979 .

[11]  S. Yakowitz.  Nonparametric Estimation of Markov Transition Functions , 1979 .

[12]  Hung T. Nguyen,et al.  Density Estimation in a Continuous-Time Stationary Markov Process , 1979 .

[13]  C. J. Stone,et al.  Optimal Rates of Convergence for Nonparametric Estimators , 1980 .

[14]  G. Banon,et al.  Recursive Estimation in Diffusion Model , 1981 .

[15]  Paul Doukhan,et al.  Estimation de la transition de probabilité d'une chaîne de Markov doëblin-récurrente. étude du cas du processus autorégressif général d'ordre 1 , 1983 .

[16]  H. T. Nguyen.  Recursive nonparametric estimation in stationary Markov processes , 1984 .

[17]  Peter Hall,et al.  Integrated Square Error Properties of Kernel Estimators of Regression Functions , 1984 .

[18]  S. Yakowitz,et al.  A comparison of kriging with nonparametric regression methods , 1985 .

[19]  Sidney Yakowitz,et al.  Markov Flow Models and the Flood Warning Problem , 1985 .