The Annals of Statistics

Nonparametric Estimation of Markov Transition Functions

Sidney Yakowitz

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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.

Article information

Ann. Statist., Volume 7, Number 3 (1979), 671-679.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M05: Markov processes: estimation
Secondary: 62G05: Estimation

Markov-chain consistent estimator nonparametric inference hydrologic time series


Yakowitz, Sidney. Nonparametric Estimation of Markov Transition Functions. Ann. Statist. 7 (1979), no. 3, 671--679. doi:10.1214/aos/1176344687.

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