Quasi maximum likelihood estimation for strongly mixing state space models and multivariate L\'evy-driven CARMA processes

We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-ime linear state space models and equidistantly observed multivariate L\'evy-driven continuoustime autoregressive moving average (MCARMA) processes. In the discrete-time setting, we prove strong consistency and asymptotic normality of the QML estimator under standard moment assumptions and a strong-mixing condition on the output process of the state space model. In the second part of the paper, we investigate probabilistic and analytical properties of equidistantly sampled continuous-time state space models and apply our results from the discrete-time setting to derive the asymptotic properties of the QML estimator of discretely recorded MCARMA processes. Under natural identifiability conditions, the estimators are again consistent and asymptotically normally distributed for any sampling frequency. We also demonstrate the practical applicability of our method through a simulation study and a data example from econometrics

A Simple Class of Bayesian Nonparametric Autoregression Models

We introduce a model for a time series of continuous outcomes, that can be expressed as fully nonparametric regression or density regression on lagged terms. The model is based on a dependent Dirichlet process prior on a family of random probability measures indexed by the lagged covariates. The approach is also extended to sequences of binary responses. We discuss implementation and applications of the models to a sequence of waiting times between eruptions of the Old Faithful Geyser, and to a dataset consisting of sequences of recurrence indicators for tumors in the bladder of several patients.MIUR 2008MK3AFZFONDECYT 1100010NIH/NCI R01CA075981Mathematic

Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes

The class of multivariate L\'{e}vy-driven autoregressive moving average (MCARMA) processes, the continuous-time analogs of the classical vector ARMA processes, is shown to be equivalent to the class of continuous-time state space models. The linear innovations of the weak ARMA process arising from sampling an MCARMA process at an equidistant grid are proved to be exponentially completely regular ($\beta$-mixing) under a mild continuity assumption on the driving L\'{e}vy process. It is verified that this continuity assumption is satisfied in most practically relevant situations, including the case where the driving L\'{e}vy process has a non-singular Gaussian component, is compound Poisson with an absolutely continuous jump size distribution or has an infinite L\'{e}vy measure admitting a density around zero.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ329 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The tail of the stationary distribution of a random coefficient AR(q) model

We investigate a stationary random coefficient autoregressive process. Using renewal type arguments tailor-made for such processes, we show that the stationary distribution has a power-law tail. When the model is normal, we show that the model is in distribution equivalent to an autoregressive process with ARCH errors. Hence, we obtain the tail behavior of any such model of arbitrary order