Bayesian estimation for first-order autoregressive model with explanatory variables

In this article, we develop a Bayesian analysis in autoregressive model with explanatory variables. When σ² is known, we consider a normal prior and give the Bayesian estimator for the regression coefficients of the model. For the case σ² is unknown, another Bayesian estimator is given for all unknown parameters under a conjugate prior. Bayesian model selection problem is also being considered under the double-exponential priors. By the convergence of ρ-mixing sequence, the consistency and asymptotic normality of the Bayesian estimators of the regression coefficients are proved. Simulation results indicate that our Bayesian estimators are not strongly dependent on the priors, and are robust.     

A general long-term aging model with different underlying activation mechanisms: Modeling,Bayesian estimation,and case influence diagnostics

In this paper we propose a general cure rate aging model. Our approach enables different underlying activation mechanisms which lead to the event of interest. The number of competing causes of the event of interest is assumed to follow a logarithmic distribution. The model is parameterized in terms of the cured fraction which is then linked to covariates. We explore the use of Markov chain Monte Carlo methods to develop a Bayesian analysis for the proposed model. Moreover, some discussions on the model selection to compare the fitted models are given, as well as case deletion influence diagnostics are developed for the joint posterior distribution based on the ψ-divergence, which has several divergence measures as particular cases, such as the Kullback–Leibler (K-L), J-distance, L₁ norm, and χ²-square divergence measures. Simulation studies are performed and experimental results are illustrated based on a real malignant melanoma data.     

Estimators of Certain Functions of the Variance of a Normal Distribution

Estimators of σ^aand log σ which are functions of Σ(x?x)²/d are considered. Besides the usual sampling theory estimators, Bayesian point estimators which are the usual measures of location of the posterior distribution are given, and in each case an exact or asymptotic expression for the divisor d is stated.     

Bayesian zero-inflated generalized Poisson regression model: estimation and case influence diagnostics

Count data with excess zeros arises in many contexts. Here our concern is to develop a Bayesian analysis for the zero-inflated generalized Poisson (ZIGP) regression model to address this problem. This model provides a useful generalization of zero-inflated Poisson model since the generalized Poisson distribution is overdispersed/underdispersed relative to Poisson. Due to the complexity of the ZIGP model, Markov chain Monte Carlo methods are used to develop a Bayesian procedure for the considered model. Additionally, some discussions on the model selection criteria are presented and a Bayesian case deletion influence diagnostics is investigated for the joint posterior distribution based on the Kullback–Leibler divergence. Finally, a simulation study and a psychological example are given to illustrate our methodology.     

Minimum phi-divergence estimators for multinomial logistic regression with complex sample design

This article develops the theoretical framework needed to study the multinomial regression model for complex sample design with pseudo-minimum phi-divergence estimators. The numerical example and the simulation study propose new estimators for the parameter of the logistic regression with overdispersed multinomial distributions for the response variables, the pseudo-minimum Cressie–Read divergence estimators, as well as new estimators for the intra-cluster correlation coefficient. The simulation study shows that the Binder’s method for the intra-cluster correlation coefficient exhibits an excellent performance when the pseudo-minimum Cressie–Read divergence estimator, with \(\lambda =\frac{2}{3}\), is plugged.     

On the Bayesian estimation and influence diagnostics for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes

The purpose of this paper is to develop a Bayesian approach for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes and presence of randomized activation mechanisms. We assume the number of competing causes of the event of interest follows a Negative Binomial (NB) distribution while the latent lifetimes are assumed to follow a Weibull distribution. Markov chain Monte Carlos (MCMC) methods are used to develop the Bayesian procedure. Model selection to compare the fitted models is discussed. Moreover, we develop case deletion influence diagnostics for the joint posterior distribution based on the ψ-divergence, which has several divergence measures as particular cases. The developed procedures are illustrated with a real data set.     

Bayesian inference and diagnostics in zero-inflated generalized power series regression model

ABSTRACT

The paper provides a Bayesian analysis for the zero-inflated regression models based on the generalized power series distribution. The approach is based on Markov chain Monte Carlo methods. The residual analysis is discussed and case-deletion influence diagnostics are developed for the joint posterior distribution, based on the ψ-divergence, which includes several divergence measures such as the Kullback–Leibler, J-distance, L₁ norm, and χ²-square in zero-inflated general power series models. The methodology is reflected in a data set collected by wildlife biologists in a state park in California.     

Bayesian inference for the Birnbaum–Saunders nonlinear regression model

We develop a Bayesian analysis for the class of Birnbaum–Saunders nonlinear regression models introduced by Lemonte and Cordeiro (Comput Stat Data Anal 53:4441–4452, 2009). This regression model, which is based on the Birnbaum–Saunders distribution (Birnbaum and Saunders in J Appl Probab 6:319–327, 1969a), has been used successfully to model fatigue failure times. We have considered a Bayesian analysis under a normal-gamma prior. Due to the complexity of the model, Markov chain Monte Carlo methods are used to develop a Bayesian procedure for the considered model. We describe tools for model determination, which include the conditional predictive ordinate, the logarithm of the pseudo-marginal likelihood and the pseudo-Bayes factor. Additionally, case deletion influence diagnostics is developed for the joint posterior distribution based on the Kullback–Leibler divergence. Two empirical applications are considered in order to illustrate the developed procedures.     

Bayesian analysis of head and neck cancer data using generalized inverse Lindley stress–strength reliability model

In this article, we present the analysis of head and neck cancer data using generalized inverse Lindley stress–strength reliability model. We propose Bayes estimators for estimating P(X > Y), when X and Y represent survival times of two groups of cancer patients observed under different therapies. The X and Y are assumed to be independent generalized inverse Lindley random variables with common shape parameter. Bayes estimators are obtained under the considerations of symmetric and asymmetric loss functions assuming independent gamma priors. Since posterior becomes complex and does not possess closed form expressions for Bayes estimators, Lindley’s approximation and Markov Chain Monte Carlo techniques are utilized for Bayesian computation. An extensive simulation experiment is carried out to compare the performances of Bayes estimators with the maximum likelihood estimators on the basis of simulated risks. Asymptotic, bootstrap, and Bayesian credible intervals are also computed for the P(X > Y).     

Estimation in noncentral distributions

This paper investigates alternatives to MIU estimators in noncentral X ² and F distributions. Two directions are pursued. In the first, a general approach for uniformly improving on MVU estimators is described and illustrated. In the second, Bayesian, procedures are characterized and illustrated as well. This effort extends earlier work of Perlman and Rasmussen and of Neff and Strawderman.     

Bayesian analysis of outlier problems using divergence measures

A Bayesian approach is presented for detecting influential observations using general divergence measures on the posterior distributions. A sampling-based approach using a Gibbs or Metropolis-within-Gibbs method is used to compute the posterior divergence measures. Four specific measures are proposed, which convey the effects of a single observation or covariate on the posterior. The technique is applied to a generalized linear model with binary response data, an overdispersed model and a nonlinear model. An asymptotic approximation using Laplace method to obtain the posterior divergence is also briefly discussed.     

Bayesian Inference on Multivariate Normal Covariance and Precision Matrices in a Star-Shaped Model with Missing Data

In this article, we study Bayesian estimation for the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in the star-shaped model with missing data. Based on a Cholesky-type decomposition of the precision matrix Ω = Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements, we develop the Jeffreys prior and a reference prior for Ψ. We then introduce a class of priors for Ψ, which includes the invariant Haar measures, Jeffreys prior, and reference prior. The posterior properties are discussed and the closed-form expressions for Bayesian estimators for the covariance matrix Σ and the precision matrix Ω are derived under the Stein loss, entropy loss, and symmetric loss. Some simulation results are given for illustration.     

LOWER BOUNDS FOR THE DIVERGENCE OF DIRECTIONAL AND AXIAL ESTIMATORS

Consider estimation of a unit vector parameter a in two classes of distributions. In the first, α is a direction. In the second, α is an axis, so that –α and α are equivalent: the aim is to obtain the projector αα^t. In each case the paper uses first principles to define measures of the divergence of such estimators and derives lower bounds for them. These bounds are computed explicitly for the Fisher-Von Mises and Scheidegger-Watson densities on the g-dimensional sphere, ω_q. In the latter case, the tightness of the bound is established by simulations.     

Pseudo‐R2 statistics under complex sampling

Model summaries based on the ratio of fitted and null likelihoods have been proposed for generalised linear models, reducing to the familiar R² coefficient of determination in the Gaussian model with identity link. In this note I show how to define the Cox–Snell and Nagelkerke summaries under arbitrary probability sampling designs, giving a design‐consistent estimator of the population model summary. It is also shown that for logistic regression models under case–control sampling the usual Cox–Snell and Nagelkerke R² are not design‐consistent, but are systematically larger than would be obtained with a cross‐sectional or cohort sample from the same population, even in settings where the weighted and unweighted logistic regression estimators are similar or identical. Implementation of the new estimators is straightforward and code is provided in R.     

Bayesian predictive modeling for Inverse Gamma-Pareto composite distribution

Inverse Gamma-Pareto composite distribution is considered as a model for heavy tailed data. The maximum likelihood (ML), smoothed empirical percentile (SM), and Bayes estimators (informative and non-informative) for the parameter θ, which is the boundary point for the supports of the two distributions are derived. A Bayesian predictive density is derived via a gamma prior for θ and the density is used to estimate risk measures. Accuracy of estimators of θ and the risk measures are assessed via simulation studies. It is shown that the informative Bayes estimator is consistently more accurate than ML, Smoothed, and the non-informative Bayes estimators.     

Small area estimation strategies for large population surveys: a comparison of design and model-based methods

Small area estimation (SAE) concerns with how to reliably estimate population quantities of interest when some areas or domains have very limited samples. This is an important issue in large population surveys, because the geographical areas or groups with only small samples or even no samples are often of interest to researchers and policy-makers. For example, large population health surveys, such as Behavioural Risk Factor Surveillance System and Ohio Mecaid Assessment Survey (OMAS), are regularly conducted for monitoring insurance coverage and healthcare utilization. Classic approaches usually provide accurate estimators at the state level or large geographical region level, but they fail to provide reliable estimators for many rural counties where the samples are sparse. Moreover, a systematic evaluation of the performances of the SAE methods in real-world setting is lacking in the literature. In this paper, we propose a Bayesian hierarchical model with constraints on the parameter space and show that it provides superior estimators for county-level adult uninsured rates in Ohio based on the 2012 OMAS data. Furthermore, we perform extensive simulation studies to compare our methods with a collection of common SAE strategies, including direct estimators, synthetic estimators, composite estimators, and Datta GS, Ghosh M, Steorts R, Maples J.'s [Bayesian benchmarking with applications to small area estimation. Test 2011;20(3):574–588] Bayesian hierarchical model-based estimators. To set a fair basis for comparison, we generate our simulation data with characteristics mimicking the real OMAS data, so that neither model-based nor design-based strategies use the true model specification. The estimators based on our proposed model are shown to outperform other estimators for small areas in both simulation study and real data analysis.     

Scale mixtures log-Birnbaum–Saunders regression models with censored data: a Bayesian approach

The main objective of this paper is to develop a full Bayesian analysis for the Birnbaum–Saunders (BS) regression model based on scale mixtures of the normal (SMN) distribution with right-censored survival data. The BS distributions based on SMN models are a very general approach for analysing lifetime data, which has as special cases the Student-t-BS, slash-BS and the contaminated normal-BS distributions, being a flexible alternative to the use of the corresponding BS distribution or any other well-known compatible model, such as the log-normal distribution. A Gibbs sample algorithm with Metropolis–Hastings algorithm is used to obtain the Bayesian estimates of the parameters. Moreover, some discussions on the model selection to compare the fitted models are given and case-deletion influence diagnostics are developed for the joint posterior distribution based on the Kullback–Leibler divergence. The newly developed procedures are illustrated on a real data set previously analysed under BS regression models.     

Nonparametric hierarchical Bayes analysis of binomial data via Bernstein polynomial priors

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators. The Canadian Journal of Statistics 40: 328–344; 2012 © 2012 Statistical Society of Canada     

A semiparametric Bayesian approach to simplex regression model with heterogeneous dispersion

ABSTRACT

Simplex regression model is often employed to analyze continuous proportion data in many studies. In this paper, we extend the assumption of a constant dispersion parameter (homogeneity) to varying dispersion parameter (heterogeneity) in Simplex regression model, and present the B-spline to approximate the smoothing unknown function within the Bayesian framework. A hybrid algorithm combining the block Gibbs sampler and the Metropolis-Hastings algorithm is presented for sampling observations from the posterior distribution. The procedures for computing model comparison criteria such as conditional predictive ordinate statistic, deviance information criterion, and averaged mean squared error are presented. Also, we develop a computationally feasible Bayesian case-deletion influence measure based on the Kullback-Leibler divergence. Several simulation studies and a real example are employed to illustrate the proposed methodologies.     

Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions

This paper deals with the Bayesian estimation of generalized exponential distribution in the proportional hazards model of random censorship under asymmetric loss functions. It is well known for the two-parameter lifetime distributions that the continuous conjugate priors for parameters do not exist; we assume independent gamma priors for the scale and the shape parameters. It is observed that the closed-form expressions for the Bayes estimators cannot be obtained; we propose Tierney–Kadane's approximation and Gibbs sampling to approximate the Bayes estimates. Monte Carlo simulation is carried out to observe the behavior of the proposed methods and one real data analysis is performed for illustration. Bayesian methods are compared with maximum likelihood and it is observed that the Bayes estimators perform better than the maximum-likelihood estimators in some cases.