Bayesian covariance matrix estimation using a mixture of decomposable graphical models

We present a Bayesian approach to estimating a covariance matrix by using a prior that is a mixture over all decomposable graphs, with the probability of each graph size specified by the user and graphs of equal size assigned equal probability. Most previous approaches assume that all graphs are equally probable. We show empirically that the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs in more efficiently estimating the covariance matrix. The prior requires knowing the number of decomposable graphs for each graph size and we give a simulation method for estimating these counts. We also present a Markov chain Monte Carlo method for estimating the posterior distribution of the covariance matrix that is much more efficient than current methods. Both the prior and the simulation method to evaluate the prior apply generally to any decomposable graphical model.     

Bayesian estimation and stochastic model specification search for dynamic survival models

Dynamic survival models are a useful extension of the popular Cox model as the effects of explanatory variables are allowed to change over time. In this paper a new auxiliary mixture sampler for Bayesian estimation of the model parameters is introduced. This sampler forms the basis of a model space MCMC method for stochastic model specification search in dynamic survival models, which involves selection of covariates to include in the model as well as specification of effects as time-varying or constant. The method is applied to two well-known data sets from the literature.     

Reliability estimation in stress–strength models: an MCMC approach

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.     

Evaluation of spatial Bayesian models—Two computational methods

Integrating a posterior function with respect to its parameters is required to compare the goodness-of-fit among Bayesian models which may have distinct priors or likelihoods. This paper is concerned with two integration methods for very high dimensional functions, using a Markovian Monte Carlo simulation or a Gaussian approximation. Numerical applications include analyses of spatial data in epidemiology and seismology.     

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.     

Basu–Dhar bivariate geometric distribution in the presence of covariates and censored data: a Bayesian approach

In this paper, we introduce classical and Bayesian approaches for the Basu–Dhar bivariate geometric distribution in the presence of covariates and censored data. This distribution is considered for the analysis of bivariate lifetime as an alternative to some existing bivariate lifetime distributions assuming continuous lifetimes as the Block and Basu or Marshall and Olkin bivariate distributions. Maximum likelihood and Bayesian estimators are presented. Two examples are considered to illustrate the proposed methodology: an example with simulated data and an example with medical bivariate lifetime data.     

Assessing the contribution of R&;D to total factor productivity—a Bayesian approach to account for heterogeneity and heteroskedasticity

This paper proposes a hierarchical Bayes estimator for a panel data random coefficient model with heteroskedasticity to assess the contribution of R&D capital to total factor productivity. Based on Hall (1993) data for 323 US firms over 1976–1990, we find that there appear to have substantial unobserved heterogeneity and heteroskedasticity across firms and industries that support the use of our Bayes inference procedure. We find much higher returns to R&D capital and a more pronounced downswing for the 1981–1985 period, followed by a more pronounced upswing than those yielded by the conventional feasible generalized least squares estimators or other estimates. The estimated elasticities of R&D capital are 0.062 for 1976–1980, 0.036 for 1981–1985 and 0.081 for 1986–1990, while the estimated elasticities of ordinary capital are much more stable over these periods.     

On estimation and influence diagnostics for log-Birnbaum–Saunders Student-t regression models: Full Bayesian analysis

The purpose of this paper is to develop a Bayesian approach for log-Birnbaum–Saunders Student-t regression models under right-censored survival data. Markov chain Monte Carlo (MCMC) methods are used to develop a Bayesian procedure for the considered model. In order to attenuate the influence of the outlying observations on the parameter estimates, we present in this paper Birnbaum–Saunders models in which a Student-t distribution is assumed to explain the cumulative damage. Also, 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 developed procedures are illustrated with a real data set.     

Nonparametric estimation of a two dimensional continuous–discrete density function by wavelets

We consider the estimation of a two dimensional continuous–discrete density function. A new methodology based on wavelets is proposed. We construct a linear wavelet estimator and a non-linear wavelet estimator based on a term-by-term thresholding. Their rates of convergence are established under the mean integrated squared error over Besov balls. In particular, we prove that our adaptive wavelet estimator attains a fast rate of convergence. A simulation study illustrates the usefulness of the proposed estimators.     

Marginal-sum and maximum-likelihood estimation in a multiplicative tariff

It is well known that, for a multiplicative tariff with independent Poisson distributed claim numbers in the different tariff cells, the maximum-likelihood estimators of the parameters satisfy the marginal-sum equations. In the present paper we show that this is also true under the more general assumption that the claim numbers of the different cells arise from the decomposition of a collective model for the whole portfolio of risks. In this general setting, the claim numbers of the different cells need not be independent and need not be Poisson distributed.     

Some applications of nonlinear and non-Gaussian state–space modelling by means of hidden Markov models

Nonlinear and non-Gaussian state–space models (SSMs) are fitted to different types of time series. The applications include homogeneous and seasonal time series, in particular earthquake counts, polio counts, rainfall occurrence data, glacial varve data and daily returns on a share. The considered SSMs comprise Poisson, Bernoulli, gamma and Student-t distributions at the observation level. Parameter estimations for the SSMs are carried out using a likelihood approximation that is obtained after discretization of the state space. The approximation can be made arbitrarily accurate, and the approximated likelihood is precisely that of a finite-state hidden Markov model (HMM). The proposed method enables us to apply standard HMM techniques. It is easy to implement and can be extended to all kinds of SSMs in a straightforward manner.     

Forecasting the Penetration of a New Product—A Bayesian Approach

We adopt a Bayesian approach to forecast the penetration of a new product into a market. We incorporate prior information from an existing product and/or management judgments into the data analysis. The penetration curve is assumed to be a nondecreasing function of time and may be under shape constraints. Markov-chain Monte Carlo methods are proposed and used to compute the Bayesian forecasts. An example on forecasting the penetration of color television using the information from black-and-white television is provided. The models considered can also be used to address the general bioassay and reliability stress-testing problems.     

Wavelets and estimation of long memory in nonstationary models: Does anything beat the exact local whittle estimator?

In this article, we analyze the performance of five estimation methods for the long memory parameter d. The goal of our article is to construct a wavelet estimate for the fractional differencing parameter in nonstationary long memory processes that dominate the well-known estimate of Shimotsu and Phillips (2005) Shimotsu, K., Phillips, P. (2005). Exact local whittle estimation of fractional integration. Annals of statistics 20:87–127. [Google Scholar]. The simulation results show that the wavelet estimation method of Lee (2005) Lee, J. (2005). Estimating memory parameter in the US inflation rate. Economics Letters 87:207–210. [Google Scholar] with several tapering techniques performs better under most cases in nonstationary long memory. The comparison is based on the empirical root mean squared error of each estimate.     

On the effect of item nonresponse on the estimation of a two–panel–waves wage equation

Summary: In this paper, we present results of the estimation of a two–panel–waves wage equation based on completely observed units and on a multiply imputed data set. In addition to the survey information, reliable income data is available from the register. These external data are used to assess the reliability of wage regressions that suffer from item nonresponse. The findings reveal marked differences between the complete case analyses and both versions of multiple imputation analyses. We argue that the results based on the multiply imputed data sets are more reliable than those based on the complete case analysis.* We would like to thank Statistics Finland for providing the data. We are also very grateful to Susanna Sandström and Marjo Pyy–Martikainen for their helpful advice using the Finnish data. Helpful comments from Joachim Winter and participants of the Workshop on Item Nonresponse and Data Quality in Large Social Surveys, Basel, October, 2003, on an earlier version of the paper are greatfully acknowledged. Further, we would like to thank three anonymous referees and the editor for helpful comments and suggestions.     

Nonparametric estimation of transition probabilities in a non-Markov illness–death model

In this paper we consider nonparametric estimation of transition probabilities for multi-state models. Specifically, we focus on the illness-death or disability model. The main novelty of the proposed estimators is that they do not rely on the Markov assumption, typically assumed to hold in a multi-state model. We investigate the asymptotic properties of the introduced estimators, such as their consistency and their convergence to a normal law. Simulations demonstrate that the new estimators may outperform Aalen–Johansen estimators (the classical nonparametric tool for estimating the transition probabilities) in non-Markov situation. An illustration through real data analysis is included.     

Accelerated test system strength models based on Birnbaum–Saunders distribution: a complete Bayesian analysis and comparison

Several models for studies related to tensile strength of materials are proposed in the literature where the size or length component has been taken to be an important factor for studying the specimens’ failure behaviour. An important model, developed on the basis of cumulative damage approach, is the three-parameter extension of the Birnbaum–Saunders fatigue model that incorporates size of the specimen as an additional variable. This model is a strong competitor of the commonly used Weibull model and stands better than the traditional models, which do not incorporate the size effect. The paper considers two such cumulative damage models, checks their compatibility with a real dataset, compares them with some of the recent toolkits, and finally recommends a model, which appears an appropriate one. Throughout the study is Bayesian based on Markov chain Monte Carlo simulation.