Journal ArticleDOI
Asymptotic Posterior Normality for Stochastic Processes Revisited
Trevor J. Sweeting,A. O. Adekola +1 more
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In this paper, a general set of conditions for asymptotic posterior normality of a posterior distribution arising from a nonhomogeneous Poisson process and birth process is presented.Abstract:
The problem of demonstrating the limiting normality of a posterior distribution arising from some stochastic process is revisited. It is shown that certain processes of practical interest are not covered by conditions currently available in the literature. In this paper we present a fairly general set of conditions for asymptotic posterior normality which cover a wide class of problems. The theory is applied to a nonhomogeneous Poisson process and birth process. case of independent and identically distributed (i.i.d.) observations, we mention Le Cam (1953, 1958), Freedman (1963), Lindley (1965), Bickel and Yahav (1969), Walker (1969), Johnson (1970), Dawid (1970), Strasser (1976) and Hartigan (1983). Markov processes, and stochastic processes more generally, are discussed in Borwanker et al (1971), Moore (1976), Heyde and Johnstone (1979), Basawa and Rao (1980) and Chen (1985). The present paper is in the spirit of Heyde and Johnstone (1979), hereafter referred to as H-J, where asymptotic posterior normality is obtained for general stochastic processes under certain conditions. These conditions however do not cover some cases of practical interest; they fail for example for certain nonhomogeneous Poisson processes which are of interest in reliability. We prove here a general result on asymptotic posterior normality which covers a wide class of problems, in particular the type of nonhomogeneous process not covered in the literature. In Section 2 we state our regularity conditions and the main result. We have specialized to the case of a single parameter for reasons of clarity and comparison with existing results. Under the regularity conditions imposed by H-J, it appears that asymptotic posterior normality holds under weaker conditions than those needed for asymptotic normality of the maximum likelihood (ML) estimator. However, a weakening of H-J's conditions, in order to cover a broader range of applications, necessitates the introduction of other conditions which also guarantee the asymptotic normality of the ML estimator. From this point of view, then, it appears that further conditions are needed for posterior normality, since nonlocal, as well asread more
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Large sample properties of posterior densities, bayesian information criterion and the likelihood principle in nonstationary time series models
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References
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Journal ArticleDOI
On the Asymptotic Behavior of Bayes' Estimates in the Discrete Case
TL;DR: In this article, it was shown that the posterior probability converges to point mass at the true parameter value among almost all sample sequences (for short, the posterior is consistent; see Definition 1) exactly for parameter values in the topological carrier of the prior.
Book
On some asymptotic properties of maximum likelihood estimates and related Bayes' estimates
Le Cam,Lucien M. (Lucien Marie) +1 more
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On the Asymptotic Behaviour of Posterior Distributions
TL;DR: In this paper, it was shown that the posterior distribution of a random sample of size n from a distribution having a density depending on a real parameter 0, and 0 having an absolutely continuous prior distribution with density ir(G) was asymptotically normal with mean equal to the maximum likelihood estimator and variance equal to a reciprocal of the second derivative of the logarithm of the likelihood function evaluated at the maximum-likelihood estimator.
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Asymptotic Expansions Associated with Posterior Distributions
TL;DR: In this paper, an extension of the investigation of Johnson (1967b) is made by giving a larger class of posterior distributions which possess asymptotic expansions having a normal distribution as a leading term.