Posterior bimodality in the balanced one-way random-effects model

Summary. Although some researchers have examined posterior multimodality for specific richly parameterized models, multimodality is not well characterized for any such model. The paper characterizes bimodality of the joint and marginal posteriors for a conjugate analysis of the balanced one-way random-effects model with a flat prior on the mean. This apparently simple model has surprisingly complex and even bizarre mode behaviour. Bimodality usually arises when the data indicate a much larger between-groups variance than does the prior. We examine an example in detail, present a graphical display for describing bimodality and use real data sets from a statistical practice to shed light on the practical relevance of bimodality for these models.     

Bayesian Conditional Mean Estimation in Log‐Normal Linear Regression Models with Finite Quadratic Expected Loss

Log‐normal linear regression models are popular in many fields of research. Bayesian estimation of the conditional mean of the dependent variable is problematic as many choices of the prior for the variance (on the log‐scale) lead to posterior distributions with no finite moments. We propose a generalized inverse Gaussian prior for this variance and derive the conditions on the prior parameters that yield posterior distributions of the conditional mean of the dependent variable with finite moments up to a pre‐specified order. The conditions depend on one of the three parameters of the suggested prior; the other two have an influence on inferences for small and medium sample sizes. A second goal of this paper is to discuss how to choose these parameters according to different criteria including the optimization of frequentist properties of posterior means.     

Prior distribution assessment for a multivariate normal distribution: an experimental study

A variety of methods of eliciting a prior distribution for a multivariate normal (MVN) distribution have recently been proposed. This paper reports an experiment in which 16 meteorologists used the methods to quantify their opinions about climatology variables. Our results compare prior models and show, in particular, that it can be better to assume the mean and variance of an MVN distribution are independent a priori, rather than to model opinion by the conjugate prior distribution. Using a proper scoring rule, different forms of assessment task are examined and alternative ways of estimating parameters are compared. To quantify opinion about means, it proved preferable to ask directly about the means rather than individual observations while, to quantify opinion about the variance matrix, it was best to ask about deviations from the mean. Further results include recommendations for the way parameters of the prior distribution are estimated.     

Conjugate analysis of multivariate normal data with incomplete observations

The authors discuss prior distributions that are conjugate to the multivariate normal likelihood when some of the observations are incomplete. They present a general class of priors for incorporating information about unidentified parameters in the covariance matrix. They analyze the special case of monotone patterns of missing data, providing an explicit recursive form for the posterior distribution resulting from a conjugate prior distribution. They develop an importance sampling and a Gibbs sampling approach to sample from a general posterior distribution and compare the two methods.     

Robust Bayesian methods in simple ANOVA models

A robust Bayesian analysis in a conjugate normal framework for the simple ANOVA model is suggested. By fixing the prior mean and varying the prior covariance matrix over a restricted class, we obtain the so-called HiFi and core region, a union and intersection of HPD regions. Based on these robust HPD regions we develop the concept of a ‘robust Bayesian judgement’ procedure. We apply this approach to the simple analysis of variance model with orthogonal designs. The example analyses the costs of an asthma medication obtained by a two-way cross-over study.     

A gamma goodness-of-fit test based on characteristic independence of the mean and coefficient of variation

Characterization theorems in probability and statistics are widely appreciated for their role in clarifying the structure of the families of probability distributions. Less well known is the role characterization theorems have as a natural, logical and effective starting point for constructing goodness-of-fit tests. The characteristic independence of the mean and variance and of the mean and the third central moment of a normal sample were used, respectively, by Lin and Mudholkar [1980. A simple test for normality against asymmetric alternatives. Biometrika 67, 455–461] and by Mudholkar et al. [2002a. Independence characterizations and testing normality against skewness-kurtosis alternatives. J. Statist. Plann. Inference 104, 485–501] for developing tests of normality. The characteristic independence of the maximum likelihood estimates of the population parameters was similarly used by Mudholkar et al. [2002b. Independence characterization and inverse Gaussian goodness-of-fit. Sankhya A 63, 362–374] to develop a test of the composite inverse Gaussian hypothesis. The gamma models are extensively used for applied research in the areas of econometrics, engineering and biomedical sciences; but there are few goodness-of-fit tests available to test if the data indeed come from a gamma population. In this paper we employ Hwang and Hu's [1999. On a characterization of the gamma distribution: the independence of the sample mean and the sample coefficient of variation. Ann. Inst. Statist. Math. 51, 749–753] characterization of the gamma population in terms of the independence of sample mean and coefficient of variation for developing such a test. The asymptotic null distribution of the proposed test statistic is obtained and empirically refined for use with samples of moderate size.     

On Properties of Predictive Priors in Linear Models

Utilizing the notion of matching predictives as in Berger and Pericchi, we show that for the conjugate family of prior distributions in the normal linear model, the symmetric Kullback-Leibler divergence between two particular predictive densities is minimized when the prior hyperparameters are taken to be those corresponding to the predictive priors proposed in Ibrahim and Laud and Laud and Ibrahim. The main application for this result is for Bayesian variable selection.     

Moments of the poly-Cauchy density with applications in estimation

Summary Simple mathematical formulae for the mean and variance of a poly-Cauchy density (proportional to a product of two Cauchy densities) are derived here and then applied to obtain Bayesian estimators for the mean of a normal population and the difference between means of two normal populations. The proposed estimators are arguably superior to the traditional estimators and to the usual Bayesian estimators, and may be highly robust.     

Non-conjugate prior distribution assessment for multivariate normal sampling

Elicitation methods are proposed for quantifying expert opinion about a multivariate normal sampling model. The natural conjugate prior family imposes a relationship between the mean vector and the covariance matrix that can portray an expert's opinion poorly. Instead we assume that opinions about the mean and the covariance are independent and suggest innovative forms of question which enable the expert to quantify separately his or her opinion about each of these parameters. Prior opinion about the mean vector is modelled by a multivariate normal distribution and about the covariance matrix by both an inverse Wishart distribution and a generalized inverse-Wishart (GIW) distribution. To construct the latter, results are developed that give insight into the GIW parameters and their interrelationships. Certain of the elicitation methods exploit unconditional assessments as fully as possible, since these can reflect an expert's beliefs more accurately than conditional assessments. Methods are illustrated through an example.     

Naive method to test the convergence of simulation and its applications in the computation of bankruptcy probability

We analyse a naive method using sample mean and sample variance to test the convergence of simulation. We find this method is valid for identically, independently distributed samples, as well as correlated samples with correlation disappearing in long period. Our simulation results on the approximation to bankruptcy probability (BP) show the naive method compares well with the Half-Width, Geweke and CUSUM methods in terms of accuracy and time cost. There are clear evidences of variance reduction from tail-distribution sampling for all convergence test methods when the true BP is very low.     

Computing some bayes estimates for the mean of a normal distribution

For sampling from a normal population with unknown mean, two families of prior densities for the mean are discussed. The corresponding posterior densities are found. A data analyst may choose a prior from these families to represent prior beliefs and then compute the corresponding Bayes estimator, using the techniques discussed.     

Estimation of variance of mean using known coefficient of variation

An optimum unbiased estimator of the variance of mean is given It is defined as a function of the mean and itscustomary unbiased variance estimator, utilizing known coefficient of variation, skewness and kurtosis of the underlying distributions. Exact results are obtained. Normal and large sample cases receive particular treatment. The proposed variance estimator is generally more efficient than the customary variance estimator; its relative efficiency becomes appreciably higher for smaller coefficient of variation, smaller sample (in the normal case at least), higher negative skewness, or higher positive skewness with sufficiently large kurtosis. The empirical findings are reassuring and supportive.     

Kullback-Leibler divergence to evaluate posterior sensitivity to different priors for autoregressive time series models

In this paper we use the Kullback-Leibler divergence to measure the distance between the posteriors of the autoregressive (AR) model coefficients, aiming to evaluate mathematically the sensitivity of the coefficients posterior to different types of priors, i.e. Jeffreys’, g, and natural conjugate priors. In addition, we evaluate the impact of the posteriors distance in Bayesian estimates of mean and variance of the model coefficients by generating a large number of Monte Carlo simulations from the posteriors. Simulation study results show that the coefficients posterior is sensitive to prior distributions, and the posteriors distance has more influence on Bayesian estimates of variance than those of mean of the model coefficients. Same results are obtained from the application to real-world time series datasets.     

A Hybrid Approximation Bayesian Test of Variance Components for Longitudinal Data

The test of variance components of possibly correlated random effects in generalized linear mixed models (GLMMs) can be used to examine if there exists heterogeneous effects. The Bayesian test with Bayes factors offers a flexible method. In this article, we focus on the performance of Bayesian tests under three reference priors and a conjugate prior: an approximate uniform shrinkage prior, modified approximate Jeffreys' prior, half-normal unit information prior and Wishart prior. To compute Bayes factors, we propose a hybrid approximation approach combining a simulated version of Laplace's method and importance sampling techniques to test the variance components in GLMMs.     

Robust Bayesian hypothesis testing in the presence of nuisance parameters

Robust Bayesian testing of point null hypotheses is considered for problems involving the presence of nuisance parameters. The robust Bayesian approach seeks answers that hold for a range of prior distributions. Three techniques for handling the nuisance parameter are studied and compared. They are (i) utilize a noninformative prior to integrate out the nuisance parameter; (ii) utilize a test statistic whose distribution does not depend on the nuisance parameter; and (iii) use a class of prior distributions for the nuisance parameter. These approaches are studied in two examples, the univariate normal model with unknown mean and variance, and a multivariate normal example.     

Comparison of testing procedures utilizing P-values and Bayes factors in some common situations

Bayesian alternatives to classical tests for several testing problems are considered. One-sided and two-sided sets of hypotheses are tested concerning an exponential parameter, a Binomial proportion, and a normal mean. Hierarchical Bayes and noninformative Bayes procedures are compared with the appropriate classical procedure, either the uniformly most powerful test or the likelihood ratio test, in the different situations. The hierarchical prior employed is the conjugate prior at the first stage with the mean being the test parameter and a noninformative prior at the second stage for the hyper parameter(s) of the first stage prior. Fair comparisons are attempted in which fair means the likelihood of making a type I error is approximately the same for the different testing procedures; once this condition is satisfied, the power of the different tests are compared, the larger the power, the better the test. This comparison is difficult in the two-sided case due to the unsurprising discrepancy between Bayesian and classical measures of evidence that have been discussed for years. The hierarchical Bayes tests appear to compete well with the typical classical test in the one-sided cases.     

Estimation of the Length Distribution of Marine Populations in the Gaussian-multinomial Setting using the Method of Moments

In this paper, the problem of estimation of the length distribution of marine populations in the Gaussian-multinomial model is considered. For the purpose of the mean and covariance parameter estimation, the method of moments estimators are developed. That is, minimum variance linear unbiased estimator for the mean frequency vector is derived and a consistent estimator for the covariance matrix of the length observations is presented. The usefulness of the proposed estimators is illustrated with an analysis of real cod length measurement data.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

Conditionally gaussian distributions and an application to kalman filtering with stochastic regressors

The work reviews theory of conditionally Gaussian distributions, especially so called theorems on normal correlation. Three theorems are given: the basic, the recursive, and the conditional theorem on normal correlation. They assume that (a,y), (a,x,y), or (a,y,z) has a Gaussian distribution, ussert that (a,y), (a,x,y), and (a,y,z), respectively, are Gaussian, and give formulas for the corresponding conditional mean vectors and variance covariance matrices. A proof is presented for the recursive and the conditional theorem.     

Bayesian analysis in multivariate regression models with conjugate priors

In this paper, we consider the full rank multivariate regression model with matrix elliptically contoured distributed errors. We formulate a conjugate prior distribution for matrix elliptical models and derive the posterior distributions of mean and scale matrices. In the sequel, some characteristics of regression matrix parameters are also proposed.