The paper discusses very general extensions to existing inflation models for discrete random variables, allowing an arbitrary set of points in the sample space to be either inflated or deflated relative to a baseline distribution. The term flation is introduced to cover either inflation or deflation of counts. Examples include one-inflated count models where the baseline distribution is zero-truncated and count models for data with a few unusual large values. The main result is that inference about the baseline distribution can be based solely on the truncated distribution which arises when the entire set of flation points is truncated. A major application of this result relates to estimating the size of a hidden target population, and examples are provided to illustrate our findings.
Many statistical models have likelihoods which are intractable: it is impossible or too expensive to compute the likelihood exactly. In such settings, a common approach is to replace the likelihood with an approximation, and proceed with inference as if the approximate likelihood were the exact likelihood. In this paper, we describe conditions on the approximate likelihood which guarantee that this naive inference with an approximate likelihood has the same first-order asymptotic properties as inference with the exact likelihood. We investigate the implications of these results for inference using a Laplace approximation to the likelihood in a simple two-level latent variable model, and using reduced dependence approximations to the likelihood in an Ising model on a lattice.
The likelihood for the parameters of a generalized linear mixed model involves an integral which may be of very high dimension. Because of this intractability, many approximations to the likelihood have been proposed, but all can fail when the model is sparse, in that there is only a small amount of information available on each random effect. The sequential reduction method described in this paper exploits the dependence structure of the posterior distribution of the random effects to reduce substantially the cost of finding an accurate approximation to the likelihood in models with sparse structure.
Laplace approximations are commonly used to approximate high-dimensional integrals in statistical applications, but the quality of such approximations as the dimension of the integral grows is not well understood. In this paper, we prove a new result on the size of the error in first-and higher-order Laplace approximations, and apply this result to investigate the quality of Laplace approximations to the likelihood in some generalized linear mixed models.
Abstract:Composite likelihoods are a class of alternatives to the full likelihood which may be used for inference in many situations where the likelihood itself is intractable. A composite likelihood estimator will be robust to certain types of model misspecification, since it may be computed without the need to specify the full distribution of the response. This potential for increased robustness has been widely discussed in recent years, and is considered a secondary motivation for the use of composite likelihood. The purpose of this paper is to show that there are some situations in which a composite likelihood estimator may actually suffer a loss of robustness compared to the maximum likelihood estimator. We demonstrate this in the case of a generalized linear mixed model under misspecification of the randomeffect distribution. As the amount of information available on each random effect increases, we show that the maximum likelihood estimator remains consistent under such misspecification, but various marginal composite likelihood estimators are inconsistent. We conclude that composite likelihood estimators cannot in general be claimed to be more robust than the maximum likelihood estimator.
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