Reference priors for a product of normal means when variances are unknown

The reference priors of Berger and Bernardo (1992) are derived for normal populations with unknown variances when the product of means is of interest. The priors are also shown to be Tibshirani's (1989) matching priors.     

The Poisson Maximum Entropy Model for Homogeneous Poisson Processes

Our main interest is parameter estimation using maximum entropy methods in the prediction of future events for Homogeneous Poisson Processes when the distribution governing the distribution of the parameters is unknown. We intend to use empirical Bayes techniques and the maximum entropy principle to model the prior information. This approach has also been motivated by the success of the gamma prior for this problem, since it is well known that the gamma maximizes Shannon entropy under appropriately chosen constraints. However, as an alternative, we propose here to apply one of the often used methods to estimate the parameters of the maximum entropy prior. It consists of moment matching, that is, maximizing the entropy subject to the constraint that the first two moments equal the empirical ones and we obtain the truncated normal distribution (truncated below at the origin) as a solution. We also use maximum likelihood estimation (MLE) methods to estimate the parameters of the truncated normal distribution for this case. These two solutions, the gamma and the truncated normal, which maximize the entropy under different constraints are tested as to their effectiveness for prediction of future events for homogeneous Poisson processes by measuring their coverage probabilities, the suitably normalized lengths of their prediction intervals and their goodness-of-fit measured by the Kullback–Leibler criterion and a discrepancy measure. The estimators obtained by these methods are compared in an extensive simulation study to each other as well as to the estimators obtained using the completely noninformative Jeffreys’ prior and the usual frequency methods. We also consider the problem of choosing between the two maximum entropy methods proposed here, that is, the gamma prior and the truncated normal prior, estimated both by matching of the first two moments and, by maximum likelihood, when faced with data and we advocate the use of the sample skewness and kurtosis. The methods are also illustrated on two examples: one concerning the occurrence of mammary tumors in laboratory animals taking part in a carcinogenicity experiment and the other, a warranty dataset from the automobile industry.     

A Note on Priors for the Multinomial Model

An “overall objective” prior proposed for the multinomial model is shown to be inadequate in the presence of zero counts. An earlier proposed reference prior for when interest is in a particular category suffers from similar problems. It is argued that there is no need to deviate from the uniform prior proposed by Jeffreys, for which links with a non-Bayesian approach, when prediction is of interest, are shown.     

Objective Bayesian analysis for a spatial model with nugget effects

The focus of this paper is objective priors for spatially correlated data with nugget effects. In addition to the Jeffreys priors and commonly used reference priors, two types of “exact” reference priors are derived based on improper marginal likelihoods. An “equivalence” theorem is developed in the sense that the expectation of any function of the score functions of the marginal likelihood function can be taken under marginal likelihoods. Interestingly, these two types of reference priors are identical.     

Objective priors for causal AR(p) with partial autocorrelations

We consider the problem of deriving formal objective priors for the causal/stationary autoregressive model of order p. We compare the frequentist behaviour of the most common default priors, namely the uniform (over the stationarity region) prior, the Jeffreys’ prior and the reference prior.     

Some extensions of zero-inflated models and Bayesian tests for them

Zero-inflated power series distribution is commonly used for modelling count data with extra zeros. Inflation at point zero has been investigated and several tests for zero inflation have been examined. However sometimes, inflation occurs at a point apart from zero. In this case, we say inflation occurs at an arbitrary point j. The j-inflation has been discussed less than zero inflation. In this paper, inflation at an arbitrary point j is studied with more details and a Bayesian test for detecting inflation at point j is presented. The Bayesian method is extended to inflation at arbitrary points i and j. The relationship between the distribution for inflation at point j, inflation at points i and j and missing value imputation is studied. It is shown how to obtain a proper estimate of the population variance if a mean-imputed missing at random data set is used. Some simulation studies are conducted and the proposed Bayesian test is applied on two real data sets.     

Reference priors for shrinkage and smoothing parameters

A reference prior and corresponding reference posteriors are derived for a basic Normal variance components model with two components. Different parameterizations are considered, in particular one in terms of a shrinkage or smoothing parameter. Earlier results for the one-way ANOVA setting are generalized and a broad range of applications of the general results is indicated. Numerical examples of application to spline smoothing are given for illustration and the results compared with other well-known techniques considered to be “non-informative” about the smoothing parameter.     

Exact calculation of minimum sample size for estimating a Poisson parameter

Abstract

We develop an exact approach for the determination of the minimum sample size for estimating a Poisson parameter such that the pre-specified levels of relative precision and confidence are guaranteed. The exact computation is made possible by reducing infinitely many evaluations of coverage probability to finitely many evaluations. The theory for supporting such a reduction is that the minimum of coverage probability with respect to the parameter in an interval is attained at a discrete set of finitely many elements. Computational mechanisms have been developed to further reduce the computational complexity. An explicit bound for the minimum sample size is established.