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.     

Estimation of a poisson mean using discrete mixtures of conjugate priors

The problem of estimating a Poisson mean is considered using incomplete prior information. The user is only able to assess two fractiles of the prior distribution. A class of mixture distributions is constructed to model this prior information; variation within this class primarily occurs in the tail region where little prior information exists. The posterior analysis using the mixture class is attractive computationally and compares favorably with the conjugate posterior analysis.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Bayesian Methods for Missing Covariates in Cure Rate Models

We propose methods for Bayesian inference for missing covariate data with a novel class of semi-parametric survival models with a cure fraction. We allow the missing covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions. We assume that the missing covariates are missing at random (MAR) throughout. We propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. The proposed class of priors are shown to be useful in recovering information on the missing covariates especially in situations where the missing data fraction is large. Properties of the proposed prior and resulting posterior distributions are examined. Also, model checking techniques are proposed for sensitivity analyses and for checking the goodness of fit of a particular model. Specifically, we extend the Conditional Predictive Ordinate (CPO) statistic to assess goodness of fit in the presence of missing covariate data. Computational techniques using the Gibbs sampler are implemented. A real data set involving a melanoma cancer clinical trial is examined to demonstrate the methodology.     

Empirical Bayes models of Poisson clinical trials and sample size determination

Bayesian methods are often used to reduce the sample sizes and/or increase the power of clinical trials. The right choice of the prior distribution is a critical step in Bayesian modeling. If the prior not completely specified, historical data may be used to estimate it. In the empirical Bayesian analysis, the resulting prior can be used to produce the posterior distribution. In this paper, we describe a Bayesian Poisson model with a conjugate Gamma prior. The parameters of Gamma distribution are estimated in the empirical Bayesian framework under two estimation schemes. The straightforward numerical search for the maximum likelihood (ML) solution using the marginal negative binomial distribution is unfeasible occasionally. We propose a simplification to the maximization procedure. The Markov Chain Monte Carlo method is used to create a set of Poisson parameters from the historical count data. These Poisson parameters are used to uniquely define the Gamma likelihood function. Easily computable approximation formulae may be used to find the ML estimations for the parameters of gamma distribution. For the sample size calculations, the ML solution is replaced by its upper confidence limit to reflect an incomplete exchangeability of historical trials as opposed to current studies. The exchangeability is measured by the confidence interval for the historical rate of the events. With this prior, the formula for the sample size calculation is completely defined. Published in 2009 by John Wiley & Sons, Ltd.     

A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution

Summary.  A useful discrete distribution (the Conway–Maxwell–Poisson distribution) is revived and its statistical and probabilistic properties are introduced and explored. This distribution is a two-parameter extension of the Poisson distribution that generalizes some well-known discrete distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions derived from these discrete distributions (i.e. the binomial and negative binomial distributions). We describe three methods for estimating the parameters of the Conway–Maxwell–Poisson distribution. The first is a fast simple weighted least squares method, which leads to estimates that are sufficiently accurate for practical purposes. The second method, using maximum likelihood, can be used to refine the initial estimates. This method requires iterations and is more computationally intensive. The third estimation method is Bayesian. Using the conjugate prior, the posterior density of the parameters of the Conway–Maxwell–Poisson distribution is easily computed. It is a flexible distribution that can account for overdispersion or underdispersion that is commonly encountered in count data. We also explore two sets of real world data demonstrating the flexibility and elegance of the Conway–Maxwell–Poisson distribution in fitting count data which do not seem to follow the Poisson distribution.     

A bivariate mixture of negative binomial distributions and its applications

Abstract

We construct a new bivariate mixture of negative binomial distributions which represents over-dispersed data more efficiently. This is an extension of a univariate mixture of beta and negative binomial distributions. Characteristics of this joint distribution are studied including conditional distributions. Some properties of the correlation coefficient are explored. We demonstrate the applicability of our proposed model by fitting to three real data sets with correlated count data. A comparison is made with some previously used models to show the effectiveness of the new model.     

Global prior distributions for the analysis of discrete graphical models

Summary We propose a new class of prior distributions for the analysis of discrete graphical models. Such a class, obtained following a conditional approach, generalizes the hyper Dirichlet distributions of Dawid and Lauritzen (1993), since it can be extended to non decomposable graphical models. The two classes are compared in terms of model selection, with an application to a medical data-set illustrating the performance of the two resulting procedures. The proposed class turns out to select simpler, more par-simonious structures.     

On robust Bayesian estimation under some asymmetric and bounded loss function

The problem of Bayesian and robust Bayesian estimation with some bounded and asymmetric loss function ABL is considered for various models. The prior distribution is not exactly specified and covers the conjugate family of prior distributions. The posterior regret, most robust and conditional Γ-minimax estimators are constructed and a preliminary comparison with square-error loss and LINEX loss is presented.     

Multivariate tail conditional expectation for scale mixtures of skew-normal distribution

In practice, a financial or actuarial data set may be a skewed or heavy-tailed and this motivates us to study a class of distribution functions in risk management theory that provide more information about these characteristics resulting in a more accurate risk analysis. In this paper, we consider a multivariate tail conditional expectation (MTCE) for multivariate scale mixtures of skew-normal (SMSN) distributions. This class of distributions contains skewed distributions and some members of this class can be used to analyse heavy-tailed data sets. We also provide a closed form for TCE in a univariate skew-normal distribution framework. Numerical examples are also provided for illustration.     

A class of rectangle-screened multivariate normal distributions and its applications

A screening problem is tackled by proposing a parametric class of distributions designed to match the behavior of the partially observed screened data. This class is obtained from the nontruncated marginal of the rectangle-truncated multivariate normal distributions. Motivations for the screened distribution as well as some of the basic properties, such as its characteristic function, are presented. These allow us a detailed exploration of other important properties that include closure property in linear transformation, in marginal and conditional operations, and in a mixture operation as well as the first two moments and some sampling distributions. Various applications of these results to the statistical modelling and data analysis are also provided.     

Robust Bayesian estimation of a bathtub-shaped distribution under progressive Type-II censoring

The bathtub-shaped failure rate function has been used for modeling the life spans of a number of electronic and mechanical products, as well as for modeling the life spans of humans, especially when some of the data are censored. This article addresses robust methods for the estimation of unknown parameters in a two-parameter distribution with a bathtub-shaped failure rate function based on progressive Type-II censored samples. Here, a class of flexible priors is considered by using the hierarchical structure of a conjugate prior distribution, and corresponding posterior distributions are obtained in a closed-form. Then, based on the square error loss function, Bayes estimators of unknown parameters are derived, which depend on hyperparameters as parameters of the conjugate prior. In order to eliminate the hyperparameters, hierarchical Bayesian estimation methods are proposed, and these proposed estimators are compared to one another based on the mean squared error, through Monte Carlo simulations for various progressively Type-II censoring schemes. Finally, a real dataset is presented for the purpose of illustration.     

Assurance in clinical trial design

Conventional clinical trial design involves considerations of power, and sample size is typically chosen to achieve a desired power conditional on a specified treatment effect. In practice, there is considerable uncertainty about what the true underlying treatment effect may be, and so power does not give a good indication of the probability that the trial will demonstrate a positive outcome. Assurance is the unconditional probability that the trial will yield a ‘positive outcome’. A positive outcome usually means a statistically significant result, according to some standard frequentist significance test. The assurance is then the prior expectation of the power, averaged over the prior distribution for the unknown true treatment effect. We argue that assurance is an important measure of the practical utility of a proposed trial, and indeed that it will often be appropriate to choose the size of the sample (and perhaps other aspects of the design) to achieve a desired assurance, rather than to achieve a desired power conditional on an assumed treatment effect. We extend the theory of assurance to two‐sided testing and equivalence trials. We also show that assurance is straightforward to compute in some simple problems of normal, binary and gamma distributed data, and that the method is not restricted to simple conjugate prior distributions for parameters. Several illustrations are given. Copyright © 2005 John Wiley & Sons, Ltd.     

Conditionally Reducible Natural Exponential Families and Enriched Conjugate Priors

Consider a standard conjugate family of prior distributions for a vector-parameter indexing an exponential family. Two distinct model parameterizations may well lead to standard conjugate families which are not consistent, i.e. one family cannot be derived from the other by the usual change-of-variable technique. This raises the problem of finding suitable parameterizations that may lead to enriched conjugate families which are more flexible than the traditional ones. The previous remark motivates the definition of a new property for an exponential family, named conditional reducibility. Features of conditionally-reducible natural exponential families are investigated thoroughly. In particular, we relate this new property to the notion of cut, and show that conditionally-reducible families admit a reparameterization in terms of a vector having likelihood-independent components. A general methodology to obtain enriched conjugate distributions for conditionally-reducible families is described in detail, generalizing previous works and more recent contributions in the area. The theory is illustrated with reference to natural exponential families having simple quadratic variance function.     

On conditional correlations defined by a class of nonlinearly truncated distributions and their applications

This article derives and studies several types of conditional correlations. The correlations are obtained by a class of two-piece scale mixture skew-normal distributions. The class is obtained by applying a set of nonlinear constraints to the bivariate scale mixture of normal distributions. The correlations of the class are invariant with respect to the choice of the scale mixing function, however, they are dependent upon the type of the nonlinear truncation. Moreover, their respective upper and lower limits are no longer 1.00 and?1.00. They are useful for the truncated data analysis, the multivariate interdependence methods (such as the principal component analysis and the factor analysis), and the random truncation modelling. Some distributional properties and the Bayesian computation of the correlations are considered when developing necessary theories and providing illustrative examples, respectively. Two applications are also given to demonstrate the usefulness of the conditional correlations in a multivariate analysis.     

Estimation and inference of the joint conditional distribution for multivariate longitudinal data using nonparametric copulas

In this paper we study estimating the joint conditional distributions of multivariate longitudinal outcomes using regression models and copulas. For the estimation of marginal models, we consider a class of time-varying transformation models and combine the two marginal models using nonparametric empirical copulas. Our models and estimation method can be applied in many situations where the conditional mean-based models are not good enough. Empirical copulas combined with time-varying transformation models may allow quite flexible modelling for the joint conditional distributions for multivariate longitudinal data. We derive the asymptotic properties for the copula-based estimators of the joint conditional distribution functions. For illustration we apply our estimation method to an epidemiological study of childhood growth and blood pressure.     

Sensitivity of a bayesian inference to prior assumptions

The sensitivity of-a Bayesian inference to prior assumptions is examined by Monte Carlo simulation for the beta-binomial conjugate family of distributions. Results for the effect on a Bayesian probability interval of the binomial parameter indicate that the Bayesian inference is for the most part quite sensitive to misspecification of the prior distribution. The magnitude of the sensitivity depends primarily on the difference of assigned means and variances from the respective means and variances of the actually-sampled prior distributions. The effect of a disparity in form between the assigned prior and actually-sampled distributions was less important for the cases tested.     

On conjugate families and Jeffreys priors for von Mises–Fisher distributions

This paper discusses characteristics of standard conjugate priors and their induced posteriors in Bayesian inference for von Mises–Fisher distributions, using either the canonical natural exponential family or the more commonly employed polar coordinate parameterizations. We analyze when standard conjugate priors as well as posteriors are proper, and investigate the Jeffreys prior for the von Mises–Fisher family. Finally, we characterize the proper distributions in the standard conjugate family of the (matrix-valued) von Mises–Fisher distributions on Stiefel manifolds.     

Variable selection in quantile regression via Gibbs sampling

Due to computational challenges and non-availability of conjugate prior distributions, Bayesian variable selection in quantile regression models is often a difficult task. In this paper, we address these two issues for quantile regression models. In particular, we develop an informative stochastic search variable selection (ISSVS) for quantile regression models that introduces an informative prior distribution. We adopt prior structures which incorporate historical data into the current data by quantifying them with a suitable prior distribution on the model parameters. This allows ISSVS to search more efficiently in the model space and choose the more likely models. In addition, a Gibbs sampler is derived to facilitate the computation of the posterior probabilities. A major advantage of ISSVS is that it avoids instability in the posterior estimates for the Gibbs sampler as well as convergence problems that may arise from choosing vague priors. Finally, the proposed methods are illustrated with both simulation and real data.     

Nonparametric binary regression using a Gaussian process prior

The article describes a nonparametric Bayesian approach to estimating the regression function for binary response data measured with multiple covariates. A multiparameter Gaussian process, after some transformation, is used as a prior on the regression function. Such a prior does not require any assumptions like monotonicity or additivity of the covariate effects. However, additivity, if desired, may be imposed through the selection of appropriate parameters of the prior. By introducing some latent variables, the conditional distributions in the posterior may be shown to be conjugate, and thus an efficient Gibbs sampler to compute the posterior distribution may be developed. A hierarchical scheme to construct a prior around a parametric family is described. A robustification technique to protect the resulting Bayes estimator against miscoded observations is also designed. A detailed simulation study is conducted to investigate the performance of the proposed methods. We also analyze some real data using the methods developed in this article.