On a Construction of Markov Models in Continuous Time

This paper studies a novel idea for constructing continuous-time stationary Markov models. The approach undertaken is based on a latent representation of the corresponding transition probabilities that conveys to appealing ways to study and simulate the dynamics of the constructed processes. Some well-known models are shown to fall within this construction shedding some light on both theoretical and applied properties. As an illustration of the capabilities of our proposal a simple estimation problem is posed.Gibbs sampler; Markov process; Stationary process

Geometric Stick-Breaking Processes for Continuous-Time Nonparametric Modeling

This paper is concerned with the construction of a continuous parameter sequence of random probability measures and its application for modeling random phenomena evolving in continuous time. At each time point we have a random probability measure which is generated by a Bayesian nonparametric hierarchical model, and the dependence structure is induced through a Wright-Fisher diffusion with mutation. The sequence is shown to be a stationary and reversible diffusion taking values on the space of probability measures. A simple estimation procedure for discretely observed data is presented and illustrated with simulated and real data sets.Bayesian non-parametric inference, continuous time dependent random measure, Markov process, measure-valued process, stationary process, stick-breaking process

Stationary models using latent structures

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Stationary mixture transition distribution (MTD) models via predictive distributions

This paper combines two ideas to construct autoregressive processes of arbitrary order. The first idea is the construction of first order stationary processes described in Pitt et al. [(2002). Constructing first order autoregressive models via latent processes. Scand. J. Statist. 29, 657-663] and the second idea is the construction of higher order processes described in Raftery [(1985). A model for high order Markov chains. J. Roy Statist. Soc. B. 47, 528-539]. The resulting models provide appealing alternatives to model non-linear and non-Gaussian time series

A density function connected with a non-negative self-decomposable random variable

The innovation random variable for a non-negative self-decomposable random variable can have a compound Poisson distribution. In this case, we provide the density function for the compounded variable. When it does not have a compound Poisson representation, there is a straightforward and easily available compound Poisson approximation for which the density function of the compounded variable is also available. These results can be used in the simulation of Ornstein-Uhlenbeck type processes with given marginal distributions. Previously, simulation of such processes used the inverse of the corresponding tail Levy measure. We show this approach corresponds to the use of an inverse cdf method of a certain distribution. With knowledge of this distribution and hence density function, the sampling procedure is open to direct sampling methods

Discussion on the paper by Caron and Fox [Sparse graphs using exchangeable random measures/ Francois Caron and Emily Fox]

Discussion of "Sparse graphs using exchangeable random measures" by F. Caron and E. Fox in view of possible extensions to multi-sample contexts and testing