In recent years the applications of multivariate Poisson models have increased, mainly because of the gradual increase in computer performance. The multivariate Poisson model used in practice is based on a common covariance term for all the pairs of variables. This is rather restrictive and does not allow for modelling the covariance structure of the data in a flexible way. In this paper we propose inference for a multivariate Poisson model with larger structure, i.e. different covariance for each pair of variables. Maximum likelihood estimation, as well as Bayesian estimation methods are proposed. Both are based on a data augmentation scheme that reflects the multivariate reduction derivation of the joint probability function. In order to enlarge the applicability of the model we allow for covariates in the specification of both the mean and the covariance parameters. Extension to models with complete structure with many multi-way covariance terms is discussed. The method is demonstrated by analyzing a real life data set.

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SummaryWe present a Bayesian forecasting methodology of discrete-time finite statespace hidden Markov models with non-constant transition matrix that depends on a set of exogenous covariates. We describe an MCMC reversible jump algorithm for predictive inference, allowing for model uncertainty regarding the set of covariates that affect the transition matrix. We apply our models to interest rates and we show that our general model formulation improves the predictive ability of standard homogeneous hidden Markov models.

Missing data are common in longitudinal studies. Likelihood-based methods ignoring the missingness mechanism are unbiased provided missingness is at random (MAR);under not-at-random missingness (MNAR), joint modeling is commonly used, often as part of sensitivity analyses. In our motivating example of modeling CD4 count trajectories during untreated HIV infection, CD4 counts are mainly censored due to treatment initiation, with the nature of this mechanism remaining debatable. Here, we evaluate the bias in the disease progression marker's change over time (slope) of a specific class of joint models, termed shared-random-effects-models (SREMs), under MAR drop-out and propose an alternative SREM model. Our proposed model relates drop-out to both the observed marker's data and the corresponding random effects, in contrast to most SREMs, which assume that the marker and the drop-out processes are independent given the random effects. We analytically calculate the asymptotic bias in two SREMs under specific MAR drop-out mechanisms, showing that the bias in marker's slope increases as the drop-out probability increases. The performance of the proposed model, and other commonly used SREMs, is evaluated under specific MAR and MNAR scenarios through simulation studies. Under MAR, the proposed model yields nearly unbiased slope estimates, whereas the other SREMs yield seriously biased estimates. Under MNAR, the proposed model estimates are approximately unbiased, whereas those from the other SREMs are moderately to heavily biased, depending on the parameterization used. The examined models are also fitted to real data and results are compared/discussed in the light of our analytical and simulation-based findings.

The forward-backward algorithm is an exact filtering algorithm which can efficiently calculate likelihoods, and which can be used to simulate from posterior distributions. Using a simple result which relates gamma random variables with different rates, we show how the forward-backward algorithm can be used to calculate the distribution of a sum of gamma random variables, and to simulate from their joint distribution given their sum. One application is to calculating the density of the time of a specific event in a Markov process, as this time is the sum of exponentially distributed interevent times. This enables us to apply the forward-backward algorithm to a range of new problems. We demonstrate our method on three problems: calculating likelihoods and simulating allele frequencies under a non-neutral population genetic model, analysing a stochastic epidemic model and simulating speciation times in phylogenetics. Copyright 2004 Royal Statistical Society.