Likelihood inference for discretely observed Markov jump processes with finite state space is investigated. The existence and uniqueness of the maximum likelihood estimator of the intensity matrix are investigated. This topic is closely related to the imbedding problem for Markov chains. It is demonstrated that the maximum likelihood estimator can be found either by the EM-algorithm or by a Markov chain Monte Carlo procedure. When the maximum likelihood estimator does not exist, an estimator can be obtained by using a penalized likelihood function or by the MCMC-procedure with a suitable prior. The theory is illustrated by a simulation study.
We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable to carry over fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach for the modelling of a real-world heavy-tailed fire insurance dataset. arXiv:1812.04139v2 [math.PR]
www.math.ku.dk/˜michaelWith a view to statistical inference for discretely observed diffusion models, we propose simple methods of simulating diffusion bridges, approximately and exactly. Diffusion bridge simulation plays a fundamental role in likelihood and Bayesian inference for diffusion processes. First a simple method of simulating approximate diffusion bridges is proposed and studied. Then these approximate bridges are used as proposal for an easily implemented Metropolis-Hastings algorithm that produces exact diffusion bridges. The new method utilizes time-reversibility properties of one-dimensional diffusions and is applicable to all one-dimensional diffusion processes with finite speed-measure. One advantage of the new approach is that simple simulation methods like the Milstein scheme can be applied to bridge simulation. Another advantage over previous bridge simulation methods is that the proposed method works well for diffusion bridges in long intervals because the computational complexity of the method is linear in the length of the interval. For ρ-mixing diffusions the approximate method is shown to be particularly accurate for long time intervals. In a simulation study, we investigate the accuracy and efficiency of the approximate method and compare it to exact simulation methods. In the study, our method provides a very good approximation to the distribution of a diffusion bridge for bridges that are likely to occur in applications to statistical inference. To illustrate the usefulness of the new method, we present an EM-algorithm for a discretely observed diffusion process.
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