Nonparametric regression on hidden \(\Phi\)-mixing variables: identifiability and consistency of a pseudo-likelihood based estimation procedure. (English) Zbl 1380.62168

Dumont, Thierry; Le Corff, Sylvain

Bernoulli 23, No. 2, 990-1021 (2017).

Summary: 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 \((X_{k})_{k\geq 0}\) is given by the process \((Y_{k})_{k\geq 0}\), 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.

MSC:

62G08	Nonparametric regression and quantile regression
60G10	Stationary stochastic processes
62G20	Asymptotic properties of nonparametric inference

Keywords:

identifiability; maximum likelihood; nonparametric estimation; state space model