Noninformative priors for the nested design

This paper considers noninformative priors for three-stage nested designs. It turns out that the noninformative prior given by Li and Stern (1997) is the one-at-a-time reference prior satisfying a second-order matching criterion when either the variance ratio or linear combinations of the means is of interest. Moreover, it is a joint probability matching prior when both the variance ratio and linear combinations of the means are of interest. These priors are compared with Jeffreys' prior in light of how accurately the coverage probabilities of Bayesian credible intervals match the corresponding frequentist coverage probabilities.     

Noninformative priors for the common mean in the bivariate normal distribution

In this paper, we consider some noninformative priors for the common mean in a bivariate normal population. We develop the first-order and second-order matching priors and reference priors. We find that the second-order matching prior is also an HPD matching prior, and matches the alternative coverage probabilities up to the second order. It turns out that derived reference priors do not satisfy a second-order matching criterion. Our simulation study indicates that the second-order matching prior performs better than the reference priors in terms of matching the target coverage probabilities in a frequentist sense. We also illustrate our results using real data.     

Approximate Bayesianity of Frequentist Confidence Intervals for a Binomial Proportion

The well-known Wilson and Agresti–Coull confidence intervals for a binomial proportion p are centered around a Bayesian estimator. Using this as a starting point, similarities between frequentist confidence intervals for proportions and Bayesian credible intervals based on low-informative priors are studied using asymptotic expansions. A Bayesian motivation for a large class of frequentist confidence intervals is provided. It is shown that the likelihood ratio interval for p approximates a Bayesian credible interval based on Kerman’s neutral noninformative conjugate prior up to O(n^{? 1}) in the confidence bounds. For the significance level α ? 0.317, the Bayesian interval based on the Jeffreys’ prior is then shown to be a compromise between the likelihood ratio and Wilson intervals. Supplementary materials for this article are available online.     

Intrinsic Bayes factor approach to a test for the power law process

The versatile new criterion called the intrinsic Bayes factor (IBF), introduced by Berger and Pericchi [J. Amer. Statist. Assoc. 91 (1996) 109–122], has made it possible to perform model selection and hypotheses testing using standard (improper) noninformative priors in a variety of situations. In this paper, we use their methodology to test several hypotheses regarding the shape parameter of the power law process, which has been widely used to model failure times of repairable systems. Assuming that we have data from the process according to the time-truncation sampling scheme, we derive the arithmetic IBFs using four default priors, including the reference and Jeffreys priors. We establish the frequentist probability matching properties of these priors. We also identify two priors that are justifiable under both time-truncation and failure-truncation schemes, so that the IBFs for both schemes can be unified. Deducing the intrinsic priors of a certain canonical form, as the time of truncation tends to infinity, we show that the arithmetic IBFs correspond asymptotically to actual Bayes factors. We also discuss the expected IBFs, which are useful with small samples. We then use these results to analyze an actual data set on the interruption times of a transmission line, summarizing our results under the default priors.     

Probability matching priors: a review

In recent years, extensive work has been done concerning the derivation of noninformative prior distributions assuring approximate frequentist validity of Bayesian inferences. This paper provides a review of matching priors obtained via quantiles andvia the distribution function. Various matching criteria are described and discussed. Emphasis is laid on a proposal of designing priors matching the true coverage probability as well as the false coverage probabilities of contiguous alternatives with the respective Bayesian counterparts. The review is not primarily meant to be a comprehensive account on the area, but, rather, to convey the main underlying ideas and point out the relationships between matching priors and other noninformative priors, such as the Jeifreys’ and the reference priors.     

The Bernstein–von Mises theorem in semiparametric competing risks models

Semiparametric Bayesian models are nowadays a popular tool in event history analysis. An important area of research concerns the investigation of frequentist properties of posterior inference. In this paper, we propose novel semiparametric Bayesian models for the analysis of competing risks data and investigate the Bernstein–von Mises theorem for differentiable functionals of model parameters. The model is specified by expressing the cause-specific hazard as the product of the conditional probability of a failure type and the overall hazard rate. We take the conditional probability as a smooth function of time and leave the cumulative overall hazard unspecified. A prior distribution is defined on the joint parameter space, which includes a beta process prior for the cumulative overall hazard. We first develop the large-sample properties of maximum likelihood estimators by giving simple sufficient conditions for them to hold. Then, we show that, under the chosen priors, the posterior distribution for any differentiable functional of interest is asymptotically equivalent to the sampling distribution derived from maximum likelihood estimation. A simulation study is provided to illustrate the coverage properties of credible intervals on cumulative incidence functions.     

Hybrid Bayesian inference on HIV viral dynamic models

Modelling of HIV dynamics in AIDS research has greatly improved our understanding of the pathogenesis of HIV-1 infection and guided for the treatment of AIDS patients and evaluation of antiretroviral therapies. Some of the model parameters may have practical meanings with prior knowledge available, but others might not have prior knowledge. Incorporating priors can improve the statistical inference. Although there have been extensive Bayesian and frequentist estimation methods for the viral dynamic models, little work has been done on making simultaneous inference about the Bayesian and frequentist parameters. In this article, we propose a hybrid Bayesian inference approach for viral dynamic nonlinear mixed-effects models using the Bayesian frequentist hybrid theory developed in Yuan [Bayesian frequentist hybrid inference, Ann. Statist. 37 (2009), pp. 2458–2501]. Compared with frequentist inference in a real example and two simulation examples, the hybrid Bayesian approach is able to improve the inference accuracy without compromising the computational load.     

Bayesian population estimation for small sample capture-recapture data using noninformative priors

One critical issue in the Bayesian approach is choosing the priors when there is not enough prior information to specify hyperparameters. Several improper noninformative priors for capture-recapture models were proposed in the literature. It is known that the Bayesian estimate can be sensitive to the choice of priors, especially when sample size is small to moderate. Yet, how to choose a noninformative prior for a given model remains a question. In this paper, as the first step, we consider the problem of estimating the population size for _Mt$M_{\mathrm{t}}$ model using noninformative priors. The _Mt$M_{\mathrm{t}}$ model has prodigious application in wildlife management, ecology, software liability, epidemiological study, census under-count, and other research areas. Four commonly used noninformative priors are considered. We find that the choice of noninformative priors depends on the number of sampling occasions only. The guidelines on the choice of noninformative priors are provided based on the simulation results. Propriety of applying improper noninformative prior is discussed. Simulation studies are developed to inspect the frequentist performance of Bayesian point and interval estimates with different noninformative priors under various population sizes, capture probabilities, and the number of sampling occasions. The simulation results show that the Bayesian approach can provide more accurate estimates of the population size than the MLE for small samples. Two real-data examples are given to illustrate the method.     

Noninformative nonparametric quantile estimation for simple random samples

For noninformative nonparametric estimation of finite population quantiles under simple random sampling, estimation based on the Polya posterior is similar to estimation based on the Bayesian approach developed by Ericson (J. Roy. Statist. Soc. Ser. B 31 (1969) 195) in that the Polya posterior distribution is the limit of Ericson's posterior distributions as the weight placed on the prior distribution diminishes. Furthermore, Polya posterior quantile estimates can be shown to be admissible under certain conditions. We demonstrate the admissibility of the sample median as an estimate of the population median under such a set of conditions. As with Ericson's Bayesian approach, Polya posterior-based interval estimates for population quantiles are asymptotically equivalent to the interval estimates obtained from standard frequentist approaches. In addition, for small to moderate sized populations, Polya posterior-based interval estimates for quantiles of a continuous characteristic of interest tend to agree with the standard frequentist interval estimates.     

Bayesian inference for a flexible class of bivariate beta distributions

Several bivariate beta distributions have been proposed in the literature. In particular, Olkin and Liu [A bivariate beta distribution. Statist Probab Lett. 2003;62(4):407–412] proposed a 3 parameter bivariate beta model which Arnold and Ng [Flexible bivariate beta distributions. J Multivariate Anal. 2011;102(8):1194–1202] extend to 5 and 8 parameter models. The 3 parameter model allows for only positive correlation, while the latter models can accommodate both positive and negative correlation. However, these come at the expense of a density that is mathematically intractable. The focus of this research is on Bayesian estimation for the 5 and 8 parameter models. Since the likelihood does not exist in closed form, we apply approximate Bayesian computation, a likelihood free approach. Simulation studies have been carried out for the 5 and 8 parameter cases under various priors and tolerance levels. We apply the 5 parameter model to a real data set by allowing the model to serve as a prior to correlated proportions of a bivariate beta binomial model. Results and comparisons are then discussed.     

An objective Bayesian analysis of the two-parameter half-logistic distribution based on progressively type-II censored samples

This paper develops an objective Bayesian analysis method for estimating unknown parameters of the half-logistic distribution when a sample is available from the progressively Type-II censoring scheme. Noninformative priors such as Jeffreys and reference priors are derived. In addition, derived priors are checked to determine whether they satisfy probability-matching criteria. The Metropolis–Hasting algorithm is applied to generate Markov chain Monte Carlo samples from these posterior density functions because marginal posterior density functions of each parameter cannot be expressed in an explicit form. Monte Carlo simulations are conducted to investigate frequentist properties of estimated models under noninformative priors. For illustration purposes, a real data set is presented, and the quality of models under noninformative priors is evaluated through posterior predictive checking.     

Approximate reference priors for Gaussian random fields

Reference priors are theoretically attractive for the analysis of geostatistical data since they enable automatic Bayesian analysis and have desirable Bayesian and frequentist properties. But their use is hindered by computational hurdles that make their application in practice challenging. In this work, we derive a new class of default priors that approximate reference priors for the parameters of some Gaussian random fields. It is based on an approximation to the integrated likelihood of the covariance parameters derived from the spectral approximation of stationary random fields. This prior depends on the structure of the mean function and the spectral density of the model evaluated at a set of spectral points associated with an auxiliary regular grid. In addition to preserving the desirable Bayesian and frequentist properties, these approximate reference priors are more stable, and their computations are much less onerous than those of exact reference priors. Unlike exact reference priors, the marginal approximate reference prior of correlation parameter is always proper, regardless of the mean function or the smoothness of the correlation function. This property has important consequences for covariance model selection. An illustration comparing default Bayesian analyses is provided with a dataset of lead pollution in Galicia, Spain.     

Model determination for the variance component model using reference priors

For the balanced variance component model when the inference concerning intraclass correlation coefficient is of interest, Bayesian analysis is often appropriate. However, the question remains is to choose the appropriate prior. In this paper, we consider testing of the intraclass correlation coefficient under a default prior specification. Berger and Bernardo's (1992) On the development of the reference prior method. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (Eds.), Bayesian Statist. Vol. 4. Oxford University Press, London, pp. 35–60 reference priors are developed and are used to obtain the intrinsic Bayes factor (Berger and Pericchi, 1996) The intrinsic Bayes factor for model selection and prediction. J. Amer. statist. Assoc. 91, 109–122 for the nested models. Influence diagnostics using intrinsic Bayes factors are also developed. Finally, one simulated data is provided which illustrates the proposed methodology with appropriate simulation based on computational formulas. Then in order to overcome the difficulty in Bayesian computation, MCMC method, such as Gibbs sampler and Metropolis–Hastings algorithm, is employed.     

Noninformative priors for linear combinations of normal means with unequal variances

For normal populations with unequal variances, we develop matching priors and reference priors for a linear combination of the means. Here, we find three second-order matching priors: a highest posterior density (HPD) matching prior, a cumulative distribution function (CDF) matching prior, and a likelihood ratio (LR) matching prior. Furthermore, we show that the reference priors are all first-order matching priors, but that they do not satisfy the second-order matching criterion that establishes the symmetry and the unimodality of the posterior under the developed priors. The results of a simulation indicate that the second-order matching prior outperforms the reference priors in terms of matching the target coverage probabilities, in a frequentist sense. Finally, we compare the Bayesian credible intervals based on the developed priors with the confidence intervals derived from real data.     

Reference priors for exponential families

Reference analysis, introduced by Bernardo (J. Roy. Statist. Soc. 41 (1979) 113) and further developed by Berger and Bernardo (On the development of reference priors (with discussion). In: J.M. Bernardo, J.O. Berger, A.P. Dawid, A.F.M. Smith (Eds.), Bayesian Statistics, Vol. 4, Clarendon Press, Oxford, pp. 35–60), has proved to be one of the most successful general methods to derive noninformative prior distributions. In practice, however, reference priors are typically difficult to obtain. In this paper we show how to find reference priors for a wide class of exponential family likelihoods.     

Bayesian comparative study on binary time series

In this paper, we consider the Bayesian analysis of binary time series with different priors, namely normal, Students' t, and Jeffreys prior, and compare the results with the frequentist methods through some simulation experiments and one real data on daily rainfall in inches at Mount Washington, NH. Among Bayesian methods, our results show that the Jeffreys prior perform better in most of the situations for both the simulation and the rainfall data. Furthermore, among weakly informative priors considered, Student's t prior with 7 degrees of freedom fits the data most adequately.     

Minimum Hellinger distance estimation for randomized play the winner design

Response-adaptive designs in clinical trials incorporate information from prior patient responses in order to assign better performing treatments to the future patients of a clinical study. An example of a response adaptive design that has received much attention in recent years is the randomized play the winner design (RPWD). Beran [1977. Minimum Hellinger distance estimates for parametric models. Ann. Statist. 5, 445–463] investigated the problem of minimum Hellinger distance procedure (MHDP) for continuous data and showed that minimum Hellinger distance estimator (MHDE) of a finite dimensional parameter is as efficient as the MLE (maximum likelihood estimator) under a true model assumption. This paper develops minimum Hellinger distance methodology for data generated using RPWD. A new algorithm using the Monte Carlo approximation to the estimating equation is proposed. Consistency and asymptotic normality of the estimators are established and the robustness and small sample performance of the estimators are illustrated using simulations. The methodology when applied to the clinical trial data conducted by Eli-Lilly and Company, brings out the treatment effect in one of the strata using the frequentist techniques compared to the Bayesian argument of Tamura et al [1994. A case study of an adaptive clinical trialin the treatment of out-patients with depressive disorder. J. Amer. Statist. Assoc. 89, 768–776].     

Progressively Type-I interval censored competing risks data for the proportional hazards family

In this article, we consider some problems of estimation and prediction when progressive Type-I interval censored competing risks data are from the proportional hazards family. The maximum likelihood estimators of the unknown parameters are obtained. Based on gamma priors, the Lindely's approximation and importance sampling methods are applied to obtain Bayesian estimators under squared error and linear–exponential loss functions. Several classical and Bayesian point predictors of censored units are provided. Also, based on given producer's and consumer's risks accepting sampling plans are considered. Finally, the simulation study is given by Monte Carlo simulations to evaluate the performances of the different methods.     

Noninformative priors for the normal variance ratio

In this paper, we consider noninformative priors for the ratio of variances in two normal populations. We develop first and second order matching priors. We find that the second order matching prior matches alternative coverage probabilities up to the second order and is also a HPD matching prior. It turns out that among the reference priors, only one-at-a-time reference prior satisfies a second order matching criterion. Our simulation study indicates that the one-at-a-time reference prior performs better than other reference priors in terms of matching the target coverage probabilities in a frequentist sense. This work is supported by Korea Research Foundation Grant (KRF-2004-002-C00041).