## Bernoulli

• Bernoulli
• Volume 23, Number 2 (2017), 990-1021.

### Nonparametric regression on hidden $\Phi$-mixing variables: Identifiability and consistency of a pseudo-likelihood based estimation procedure

#### Abstract

This paper outlines a new nonparametric estimation procedure for unobserved $\Phi$-mixing processes. It is assumed that the only information on the stationary hidden states $(X_{k})_{k\ge0}$ is given by the process $(Y_{k})_{k\ge0}$, where $Y_{k}$ is a noisy observation of $f_{\star}(X_{k})$. The paper introduces a maximum pseudo-likelihood procedure to estimate the function $f_{\star}$ and the distribution $\nu_{b,\star}$ of $(X_{0},\ldots,X_{b-1})$ using blocks of observations of length $b$. The identifiability of the model is studied in the particular cases $b=1$ and $b=2$ and the consistency of the estimators of $f_{\star}$ and of $\nu_{b,\star}$ as the number of observations grows to infinity is established.

#### Article information

Source
Bernoulli, Volume 23, Number 2 (2017), 990-1021.

Dates
Revised: July 2015
First available in Project Euclid: 4 February 2017

https://projecteuclid.org/euclid.bj/1486177390

Digital Object Identifier
doi:10.3150/15-BEJ767

Mathematical Reviews number (MathSciNet)
MR3606757

Zentralblatt MATH identifier
1380.62168

#### Citation

Dumont, Thierry; Le Corff, Sylvain. Nonparametric regression on hidden $\Phi$-mixing variables: Identifiability and consistency of a pseudo-likelihood based estimation procedure. Bernoulli 23 (2017), no. 2, 990--1021. doi:10.3150/15-BEJ767. https://projecteuclid.org/euclid.bj/1486177390

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