Bayesian analysis of repairable systems with modulated power law process

As a compromise between nonhomogeneous Poisson process and renewal process, the modulated power law process is more appropriate to model the failures of repairable systems. In this article, objective Bayesian methods are proposed to analyze the modulated power law process. Seven reference priors, one of which is also the Jeffreys prior, are derived. However, only four of them are taken into consideration because of their practicality. Propriety of the posterior densities considering the four reference priors is proved. Predictive distribution of the future failure time is obtained additionally. For the purpose of comparison, the simulation work and real data analysis are carried out based on both the objective Bayesian and maximum likelihood approaches, which show that the objective Bayesian estimation and prediction have much better statistical properties in a frequentist context, and outperforms the maximum likelihood method even with small or moderate sample sizes.     

Bayesian reference analysis for Gaussian Markov random fields

Gaussian Markov random fields (GMRF) are important families of distributions for the modeling of spatial data and have been extensively used in different areas of spatial statistics such as disease mapping, image analysis and remote sensing. GMRFs have been used for the modeling of spatial data, both as models for the sampling distribution of the observed data and as models for the prior of latent processes/random effects; we consider mainly the former use of GMRFs. We study a large class of GMRF models that includes several models previously proposed in the literature. An objective Bayesian analysis is presented for the parameters of the above class of GMRFs, where explicit expressions for the Jeffreys (two versions) and reference priors are derived, and for each of these priors results on posterior propriety of the model parameters are established. We describe a simple MCMC algorithm for sampling from the posterior distribution of the model parameters, and study frequentist properties of the Bayesian inferences resulting from the use of these automatic priors. Finally, we illustrate the use of the proposed GMRF model and reference prior for studying the spatial variability of lip cancer cases in the districts of Scotland over the period 1975-1980.     

Objective Bayesian analysis for CAR models

Objective priors, especially reference priors, have been studied extensively for spatial data in the last decade. In this paper, we study objective priors for a CAR model. In particular, the properties of the reference prior and the corresponding posterior are studied. Furthermore, we show that the frequentist coverage probabilities of posterior credible intervals depend only on the spatial dependence parameter $\rho $ , and not on the regression coefficient or the error variance. Based on the simulation study for comparing the reference and Jeffreys priors, the performance of two reference priors is similar and better than the Jeffreys priors. One spatial dataset is used for illustration.     

Objective Bayesian analysis for the accelerated degradation model based on the inverse Gaussian process

The inverse Gaussian process is an attractive stochastic process to model monotone degradation paths in degradation analysis. In this paper, we propose an objective Bayesian method to analyze the accelerated degradation model based on the inverse Gaussian process. Noninformative priors including the Jeffreys prior and reference priors are derived, and the propriety of the posteriors under each prior is validated. A simulation study is carried out to investigate the performance of the approach compared with the maximum likelihood method and the Bootstrap method. Numerical results show that the proposed method has better performance in terms of the mean squared error and the frequentist coverage probability. The reference prior ${\pi }_{R_2}$ is recommended to use in practice. Finally, the Bayesian approach is applied to a real data.     

Reference Bayesian analysis of inverse Gaussian degradation process

In this paper, objective Bayesian method is applied to analyze degradation model based on the inverse Gaussian process. Noninformative priors (Jefferys prior and two reference priors) for model parameters are obtained and their properties are discussed. Moreover, we propose a class of modified reference priors to remedy weaknesses of the usual reference priors and show that the modified reference priors not only have proper posterior distributions but also have probability matching properties for model parameters. Gibbs sampling algorithms for Bayesian inference based on the Jefferys prior and the modified reference priors are studied. Simulations are conducted to compare the objective Bayesian estimates with the maximum likelihood estimates and subjective Bayesian estimates and shows better performance of the objective method than the other two estimates especially for the case of small sample size. Finally, two real data examples are analyzed for illustration.     

Bayesian priors in sequential binomial design

The status of sequential analysis in Bayesian inference is revisited. The information on the experimental design, including the stopping rule, is one part of the evidence, prior to the sampling. Consequently this information must be incorporated in the prior distribution. This approach allows to relax the likelihood principle when appropriate. It is illustrated in the case of successive Binomial trials. Using Jeffreys' rule, a prior based on the Fisher information and conditional on the design characteristics is derived. The corrected Jeffreys prior, which involves a new distribution called Beta-J, extends the classical Jeffreys priors for the Binomial and Pascal sampling models to more general stopping rules. As an illustration, we show that the correction induced on the posterior is proportional to the bias induced by the stopping rule on the maximum likelihood estimator. To cite this article: P. Bunouf, B. Lecoutre, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

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Step-stress accelerated degradation test (SSADT) is a useful tool for assessing the lifetime distribution of highly reliable products when the available test items are very few. In this paper, we discuss multiple-steps step-stress accelerated degradation models based on Wiener process, and we apply the objective Bayesian method for such analytically intractable models to obtain the noninformative priors (Jefferys prior and two Reference priors). Moreover, we show that their posterior distributions are proper, and we propose Gibbs sampling algorithms for the Bayesian inference based on the Jefferys prior and two Reference priors. Finally, we present some simulation studies to compare the objective Bayesian estimates with the other Bayesian estimate and the maximum likelihood estimates (MLEs). Simulation results demonstrate the superiority of objective Bayesian analysis method.     

Priors for Bayesian adaptive spline smoothing

Adaptive smoothing has been proposed for curve-fitting problems where the underlying function is spatially inhomogeneous. Two Bayesian adaptive smoothing models, Bayesian adaptive smoothing splines on a lattice and Bayesian adaptive P-splines, are studied in this paper. Estimation is fully Bayesian and carried out by efficient Gibbs sampling. Choice of prior is critical in any Bayesian non-parametric regression method. We use objective priors on the first level parameters where feasible, specifically independent Jeffreys priors (right Haar priors) on the implied base linear model and error variance, and we derive sufficient conditions on higher level components to ensure that the posterior is proper. Through simulation, we demonstrate that the common practice of approximating improper priors by proper but diffuse priors may lead to invalid inference, and we show how appropriate choices of proper but only weakly informative priors yields satisfactory inference.     

A class of proper priors for Bayesian simultaneous prediction of independent Poisson observables

Simultaneous prediction and parameter inference for the independent Poisson observables model are considered. A class of proper prior distributions for Poisson means is introduced. Bayesian predictive densities and estimators based on priors in the introduced class dominate the Bayesian predictive density and estimator based on the Jeffreys prior under Kullback-Leibler loss.     

Objective bayesian analysis of the Yule–Simon distribution with applications

The Yule–Simon distribution is usually employed in the analysis of frequency data. As the Bayesian literature, so far, has ignored this distribution, here we show the derivation of two objective priors for the parameter of the Yule–Simon distribution. In particular, we discuss the Jeffreys prior and a loss-based prior, which has recently appeared in the literature. We illustrate the performance of the derived priors through a simulation study and the analysis of real datasets.     

Estimation of a Multivariate Normal Covariance Matrix with Staircase Pattern Data

In this paper, we study the problem of estimating a multivariate normal covariance matrix with staircase pattern data. Two kinds of parameterizations in terms of the covariance matrix are used. One is Cholesky decomposition and another is Bartlett decomposition. Based on Cholesky decomposition of the covariance matrix, the closed form of the maximum likelihood estimator (MLE) of the covariance matrix is given. Using Bayesian method, we prove that the best equivariant estimator of the covariance matrix with respect to the special group related to Cholesky decomposition uniquely exists under the Stein loss. Consequently, the MLE of the covariance matrix is inadmissible under the Stein loss. Our method can also be applied to other invariant loss functions like the entropy loss and the symmetric loss. In addition, based on Bartlett decomposition of the covariance matrix, the Jeffreys prior and the reference prior of the covariance matrix with staircase pattern data are also obtained. Our reference prior is different from Berger and Yang’s reference prior. Interestingly, the Jeffreys prior with staircase pattern data is the same as that with complete data. The posterior properties are also investigated. Some simulation results are given for illustration.     

Equiaffine structures on statistical manifolds and Bayesian statistics

Relations between equiaffine geometry and Bayesian statistics are studied. A prior distribution in Bayesian statistics is regarded as a volume form on a statistical manifold. Applying equiaffine geometry to Bayesian statistics, the relation between alpha-parallel priors and the Jeffreys prior is given. As geometric results, conditions for a statistical submanifold to have an equiaffine structure are also given.     

Bayesian Analysis for Random Coefficient Regression Models Using Noninformative Priors

We apply Bayesian approach, through noninformative priors, to analyze a Random Coefficient Regression (RCR) model. The Fisher information matrix, the Jeffreys prior and reference priors are derived for this model. Then, we prove that the corresponding posteriors are proper when the number of full rank design matrices are greater than or equal to twice the number of regression coefficient parameters plus 1 and that the posterior means for all parameters exist if one more additional full rank design matrix is available. A hybrid Markov chain sampling scheme is developed for computing the Bayesian estimators for parameters of interest. A small-scale simulation study is conducted for comparing the performance of different noninformative priors. A real data example is also provided and the data are analyzed by a non-Bayesian method as well as Bayesian methods with noninformative priors.     

Computation of reference Bayesian inference for variance components in longitudinal studies

Generalized linear mixed models (GLMMs) have been applied widely in the analysis of longitudinal data. This model confers two important advantages, namely, the flexibility to include random effects and the ability to make inference about complex covariances. In practice, however, the inference of variance components can be a difficult task due to the complexity of the model itself and the dimensionality of the covariance matrix of random effects. Here we first discuss for GLMMs the relation between Bayesian posterior estimates and penalized quasi-likelihood (PQL) estimates, based on the generalization of Harville’s result for general linear models. Next, we perform fully Bayesian analyses for the random covariance matrix using three different reference priors, two with Jeffreys’ priors derived from approximate likelihoods and one with the approximate uniform shrinkage prior. Computations are carried out via the combination of asymptotic approximations and Markov chain Monte Carlo methods. Under the criterion of the squared Euclidean norm, we compare the performances of Bayesian estimates of variance components with that of PQL estimates when the responses are non-normal, and with that of the restricted maximum likelihood (REML) estimates when data are assumed normal. Three applications and simulations of binary, normal, and count responses with multiple random effects and of small sample sizes are illustrated. The analyses examine the differences in estimation performance when the covariance structure is complex, and demonstrate the equivalence between PQL and the posterior modes when the former can be derived. The results also show that the Bayesian approach, particularly under the approximate Jeffreys’ priors, outperforms other procedures.     

The Sensitivity of the Number of Clusters in a Gaussian Mixture Model to Prior Distributions

One of the main advantages of Bayesian approaches is that they offer principled methods of inference in models of varying dimensionality and of models of infinite dimensionality. What is less widely appreciated is how the model inference is sensitive to prior distributions and therefore how priors should be set for real problems. In this paper prior sensitivity is considered with respect to the problem of inference in Gaussian mixture models. Two distinct Bayesian approaches have been proposed. The first is to use Bayesian model selection based upon the marginal likelihood; the second is to use an infinite mixture model which ‘side steps’ model selection. Explanations for the prior sensitivity are given in order to give practitioners guidance in setting prior distributions. In particular the use of conditionally conjugate prior distributions instead of purely conjugate prior distributions are advocated as a method for investigating prior sensitivity of the mean and variance individually.     

Bayesian multivariate spatial models for roadway traffic crash mapping

We consider several Bayesian multivariate spatial models for estimating the crash rates from different kinds of crashes. Multivariate conditional autoregressive (CAR) models are considered to account for the spatial effect. The models considered are fully Bayesian. A general theorem for each case is proved to ensure posterior propriety under noninformative priors. The different models are compared according to some Bayesian criterion. Markov chain Monte Carlo (MCMC) is used for computation. We illustrate these methods with Texas Crash Data.     

Monte Carlo approximation through Gibbs output in generalized linear mixed models

Geyer (J. Roy. Statist. Soc. 56 (1994) 291) proposed Monte Carlo method to approximate the whole likelihood function. His method is limited to choosing a proper reference point. We attempt to improve the method by assigning some prior information to the parameters and using the Gibbs output to evaluate the marginal likelihood and its derivatives through a Monte Carlo approximation. Vague priors are assigned to the parameters as well as the random effects within the Bayesian framework to represent a non-informative setting. Then the maximum likelihood estimates are obtained through the Newton Raphson method. Thus, out method serves as a bridge between Bayesian and classical approaches. The method is illustrated by analyzing the famous salamander mating data by generalized linear mixed models.     

Objective Bayesian analysis for competing risks model with Wiener degradation phenomena and catastrophic failures

In this paper, the objective Bayesian method is applied to investigate the competing risks model involving both catastrophic and degradation failures. By modeling soft failure as the Wiener degradation process, and hard failures as a Weibull distribution, we obtain the noninformative priors (Jefferys prior and two reference priors) for the parameters. Moreover, we show that their posterior distributions have good properties and we propose Gibbs sampling algorithms for the Bayesian inference based on the Jefferys prior and two reference priors. Some simulation studies are conducted to illustrate the superiority of objective Bayesian method. Finally, we apply our methods to two real data examples and compare the objective Bayesian estimates with the other estimates.     

Efficient Sampling for Gaussian Linear Regression With Arbitrary Priors

This article develops a slice sampler for Bayesian linear regression models with arbitrary priors. The new sampler has two advantages over current approaches. One, it is faster than many custom implementations that rely on auxiliary latent variables, if the number of regressors is large. Two, it can be used with any prior with a density function that can be evaluated up to a normalizing constant, making it ideal for investigating the properties of new shrinkage priors without having to develop custom sampling algorithms. The new sampler takes advantage of the special structure of the linear regression likelihood, allowing it to produce better effective sample size per second than common alternative approaches.     

Bayesian Statistical Analysis and Application of Masked Data based on Pareto Distribution

In this paper Bayesian statistical analysis of masked data is considered based on the Pareto distribution. The likelihood function is simplified by introducing auxiliary variables, which describe the causes of failure. Three Bayesian approaches (Bayes using subjective priors, hierarchical Bayes and empirical Bayes) are utilized to estimate the parameters, and we compare these methods by analyzing a real data. Finally we discuss the method of avoiding the choice of the hyperparameters in the prior distributions.