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May, 1979 Nonparametric Estimation of Markov Transition Functions
Sidney Yakowitz
Ann. Statist. 7(3): 671-679 (May, 1979). DOI: 10.1214/aos/1176344687


Let $\{X_n\}$ be a Markov chain having a stationary transition function and assume that the state set is an arbitrary set in a Euclidean space. The state transition law of the chain is given by a function $F(y|x) = P\lbrack X_{n+1} \leqslant y|X_n = x\rbrack$, which is assumed defined and continuous for all $x$. In this paper we give a statistical procedure for determining a function $F_n(y\mid x)$ on the basis of the sample $\{X_j\}^n_{j=1}, n = 1, 2,\cdots,$ and prove that if the chain is irreducible, aperiodic, and possesses a limiting distribution $\pi$, then with probability 1, $\sup_y|F_n(y|x) - F(y|x)| \rightarrow_n0$ for every $x$ such that any open sphere containing $x$ has positive $\pi$ probability. This result improves upon a study by Roussas which gives only weak convergence. We demonstrate that a certain clustering algorithm is useful for obtaining efficient versions of our estimates. The potential value of our methods is illustrated by computer studies using simulated data.


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Sidney Yakowitz. "Nonparametric Estimation of Markov Transition Functions." Ann. Statist. 7 (3) 671 - 679, May, 1979.


Published: May, 1979
First available in Project Euclid: 12 April 2007

zbMATH: 0407.62060
MathSciNet: MR527501
Digital Object Identifier: 10.1214/aos/1176344687

Primary: 62M05
Secondary: 62G05

Keywords: consistent estimator , hydrologic time series , Markov-chain , nonparametric inference

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 3 • May, 1979
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