The Annals of Statistics

Nonparametric Estimation of Markov Transition Functions

Sidney Yakowitz

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Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176344687

Digital Object Identifier
doi:10.1214/aos/1176344687

Mathematical Reviews number (MathSciNet)
MR527501

Zentralblatt MATH identifier
0407.62060

JSTOR
links.jstor.org

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

Keywords
Markov-chain consistent estimator nonparametric inference hydrologic time series

Citation

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


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