A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data

Frequently in the analysis of survival data, survival times within the same group are correlated due to unobserved co-variates. One way these co-variates can be included in the model is as frailties. These frailty random block effects generate dependency between the survival times of the individuals which are conditionally independent given the frailty. Using a conditional proportional hazards model, in conjunction with the frailty, a whole new family of models is introduced. By considering a gamma frailty model, often the issue is to find an appropriate model for the baseline hazard function. In this paper a flexible baseline hazard model based on a correlated prior process is proposed and is compared with a standard Weibull model. Several model diagnostics methods are developed and model comparison is made using recently developed Bayesian model selection criteria. The above methodologies are applied to the McGilchrist and Aisbett (1991) kidney infection data and the analysis is performed using Markov Chain Monte Carlo methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Protection against outliers in empirical bayes estimation

The proven optimality properties of empirical Bayes estimators and their documented successful performance in practice have made them popular. Although many statisticians have used these estimators since the landmark paper of James and Stein (1961), relatively few have proposed techniques for protecting them from the effects of outlying observations or outlying parameters. One notable series of studies in protection against outlying parameters was conducted by Efron and Morris (1971, 1972, 1975). In the fully Bayesian case, a general discussion on robust procedures can be found in Berger (1984, 1985). Here we implement and evaluate a different approach for outlier protection in a random-effects model which is based on appropriate specification of the prior distribution. When unusual parameters are present, we estimate the prior as a step function, as suggested by Laird and Louis (1987). This procedure is evaluated empirically, using a number of simulated data sets to compare the effects of the step-function prior with those of the normal and Laplace priors on the prediction of small-area proportions.     

A Comparison of the Cost-efficiencies of the Sequential, Group-sequential, and Variable-sample-size-sequential Probability Ratio Tests

Wald and Wolfowitz (1948) have shown that the Sequential Probability Ratio Test (SPRT) for deciding between two simple hypotheses is, under very restrictive conditions, optimal in three attractive senses. First, it can be a Bayes-optimal rule. Second, of all level α tests having the same power, the test with the smallest joint-expected number of observations is the SPRT, where this expectation is taken jointly with respect to both data and prior over the two hypotheses. Third, the level α test needing the fewest conditional-expected number of observat ions is the SPRT, where this expectation is now taken with respect to the data conditional on either hypothesis being true. Principal among the strong restrictions is that sampling can proceed only in a one-at-a-time manner. In this paper, we relax some of the conditions and show that there are sequential procedures that strictly dominate the SPRT in all three senses. We conclude that the third type of optimality occurs rarely and that decision-makers are better served by looking for sequential procedures that possess the first two types of optimality. By relaxing the one-at-a-time sampling restriction, we obtain optimal (in the first two senses) variable-s ample-size- sequential probability ratio tests.     

Bayesian robustness in bidimensional models: Prior independence

When θ is a multidimensional parameter, the issue of prior dependence or independence of coordinates is a serious concern. This is especially true in robust Bayesian analysis; Lavine et al. (J. Amer. Statist. Assoc.86, 964–971 (1991)) show that allowing a wide range of prior dependencies among coordinates can result in near vacuous conclusions. It is sometimes possible, however, to make confidently the judgement that the coordinates of θ are independent a priori and, when this can be done, robust Bayesian conclusions improve dramatically. In this paper, it is shown how to incorporate the independence assumption into robust Bayesian analysis involving -contamination and density band classes of priors. Attention is restricted to the case θ = (θ₁, θ₂) for clarity, although the ideas generalize.     

Bayesian reconstruction of the missing failure times in exponential distribution

In this paper, we consider the situations in which a middle part of the sample is lost or removed from the experiment. The left-censored sample and Type-II right-censored sample are special cases of this scheme. Point and interval reconstructors for lost failure times are obtained, while the underlying distribution is exponential. To illustrate the performance of the proposed procedures, two numerical examples based on the real data sets are presented. In the case of two-parameter, a simulation study has been done and also the sensitivity of the results to deviations from the true distribution is investigated.     

A procedure to select a ML-II prior in a multivariate normal case

Berger (1985) derived a procedure to select a maximum likelihood II prior distribution. In this paper a method is suggested to construct such a prior distribution from a multivariate ε-contamination class of distributions. The method is illustrated by the conetruction of a ML-II prior in the multivariate normal case.     

Comparison of proportions with asymmetric prior information

In this paper a Bayesian model is developed for comparing two binomial proportions. A two stage hierarchical prior distribution is used to represent prior dependence. Prior exchangeability and independence are shown to be but special cases. The relevant distributions have to be computed numerically and some examples are presented.     

Real-Time Forecasting With a Mixed-Frequency VAR

This article develops a vector autoregression (VAR) for time series which are observed at mixed frequencies—quarterly and monthly. The model is cast in state-space form and estimated with Bayesian methods under a Minnesota-style prior. We show how to evaluate the marginal data density to implement a data-driven hyperparameter selection. Using a real-time dataset, we evaluate forecasts from the mixed-frequency VAR and compare them to standard quarterly frequency VAR and to forecasts from MIDAS regressions. We document the extent to which information that becomes available within the quarter improves the forecasts in real time. This article has online supplementary materials.     

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.     

Nonparametric Bayesian analysis for multi-site hidden Markov model

The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in a variety of applications including pattern recognition. Unlike typical mixture models, hidden Markov states can represent the heterogeneity in data and it can be extended to a multivariate case using a hierarchical Bayesian approach. This article provides a nonparametric Bayesian modeling approach to the multi-site HMM by considering stick-breaking priors for each row of an infinite state transition matrix. This extension has many advantages over a parametric HMM. For example, it can provide more flexible information for identifying the structure of the HMM than parametric HMM analysis, such as the number of states in HMM. We exploit a simulation example and a real dataset to evaluate the proposed approach.