A Nonparametric Model for Stationary Time Series

Abstract

Stationary processes are a natural choice as statistical models for time series data, owing to their good estimating properties. In practice, however, alternative models are often proposed that sacrifice stationarity in favour of the greater modelling flexibility required by many real-life applications. We present a family of time-homogeneous Markov processes with nonparametric stationary densities, which retain the desirable statistical properties for inference, while achieving substantial modelling flexibility, matching those achievable with certain non-stationary models. A latent extension of the model enables exact inference through a trans-dimensional Markov chain Monte Carlo method. Numerical illustrations are presented.

Bayesian consistency for a nonparametric stationary Markov model

“…In this section we introduce the nonparametric Markov model studied in [2]. We provide three specific examples including the one considered in [2]. The other two examples possess interesting statistical properties.…”

“…be the density of a bivariate mixture. As explained in the introduction, the notation f P is used to denote the conditional density 2) and the marginal…”

“…Since the conditional density (2.2) is parameterized by a mixing distribution P ∈ P, for a Bayesian analysis of the Markov chain X, we only need to choose an appropriate parametric family K and put a prior on the space of mixing distribution P. The only requirement for K and P is that f P (·|·) ∈ F for every P ∈ P. As mentioned in [2], f P (·|·) ∈ F if the two marginals of f P (·, ·) are identical. Based on this idea, they proposed to use bivariate normal kernels with the same mean and variance parameters; see Section 2.1.…”

“…Markov models) is now widely routine following the development of mixture transition densities which are tractable and describe stationary densities which are also mixture models. See for example [2] and the references within that paper. While there is now a substantial literature on Bayesian nonparametric posterior consistency for i.i.d.…”

“…In this paper, we consider posterior consistency for a Markov model with a novel class of nonparametric prior, which is an extension of [2]. In this model, the transition density is parameterized by a mixing distribution function, so a metric between mixing distributions can be used to construct neighborhoods of a transition density.…”