## The Annals of Statistics

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

### Nonparametric Estimation of Markov Transition Functions

#### 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