Bayesian inference for comparing two Poisson rates using data subject to underreporting and validation data

We derive a new Bayesian credible interval estimator for comparing two Poisson rates when counts are underreported and an additional validation data set is available. We provide a closed-form posterior density for the difference between the two rates that yields insightful information on which prior parameters influence the posterior the most. We also apply the new interval estimator to a real-data example, investigate the performance of the credible interval, and examine the impact of informative priors on the rate difference posterior via Monte Carlo simulations.     

Intrinsic Bayesian inference on a Poisson rate and on the ratio of two Poisson rates

Our purpose is to explore the intrinsic Bayesian inference on the rate of a Poisson distribution and on the ratio of the rates of two independent Poisson distributions, with the natural conjugate family of priors in the first case and the semi-conjugate family of priors defined by Laurent and Legrand (2011) in the second case. Intrinsic Bayesian inference is derived from the Bayesian decision theory framework based on the intrinsic discrepancy loss function. We cover in particular the case of some objective Bayesian procedures suggested by Bernardo when considering reference priors.     

A matching prior for the product of normal means based on the modified profile likelihood

In this paper, we develop a matching prior for the product of means in several normal distributions with unrestricted means and unknown variances. For this problem, properly assigning priors for the product of normal means has been issued because of the presence of nuisance parameters. Matching priors, which are priors matching the posterior probabilities of certain regions with their frequentist coverage probabilities, are commonly used but difficult to derive in this problem. We developed the first order probability matching priors for this problem; however, the developed matching priors are unproper. Thus, we apply an alternative method and derive a matching prior based on a modification of the profile likelihood. Simulation studies show that the derived matching prior performs better than the uniform prior and Jeffreys’ prior in meeting the target coverage probabilities, and meets well the target coverage probabilities even for the small sample sizes. In addition, to evaluate the validity of the proposed matching prior, Bayesian credible interval for the product of normal means using the matching prior is compared to Bayesian credible intervals using the uniform prior and Jeffrey’s prior, and the confidence interval using the method of Yfantis and Flatman.     

Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions

The lognormal distribution is currently used extensively to describe the distribution of positive random variables. This is especially the case with data pertaining to occupational health and other biological data. One particular application of the data is statistical inference with regards to the mean of the data. Other authors, namely Zou et al. (2009), have proposed procedures involving the so-called “method of variance estimates recovery” (MOVER), while an alternative approach based on simulation is the so-called generalized confidence interval, discussed by Krishnamoorthy and Mathew (2003). In this paper we compare the performance of the MOVER-based confidence interval estimates and the generalized confidence interval procedure to coverage of credibility intervals obtained using Bayesian methodology using a variety of different prior distributions to estimate the appropriateness of each. An extensive simulation study is conducted to evaluate the coverage accuracy and interval width of the proposed methods. For the Bayesian approach both the equal-tail and highest posterior density (HPD) credibility intervals are presented. Various prior distributions (Independence Jeffreys' prior, Jeffreys'-Rule prior, namely, the square root of the determinant of the Fisher Information matrix, reference and probability-matching priors) are evaluated and compared to determine which give the best coverage with the most efficient interval width. The simulation studies show that the constructed Bayesian confidence intervals have satisfying coverage probabilities and in some cases outperform the MOVER and generalized confidence interval results. The Bayesian inference procedures (hypothesis tests and confidence intervals) are also extended to the difference between two lognormal means as well as to the case of zero-valued observations and confidence intervals for the lognormal variance. In the last section of this paper the bivariate lognormal distribution is discussed and Bayesian confidence intervals are obtained for the difference between two correlated lognormal means as well as for the ratio of lognormal variances, using nine different priors.     

Bayesian analysis for incomplete multi-way contingency tables with nonignorable nonresponse

We propose Bayesian methods with five types of priors to estimate cell probabilities in an incomplete multi-way contingency table under nonignorable nonresponse. In this situation, the maximum likelihood (ML) estimates often fall in the boundary solution, causing the ML estimates to become unstable. To deal with such a multi-way table, we present an EM algorithm which generalizes the previous algorithm used for incomplete one-way tables. Three of the five types of priors were previously introduced while the other two are newly proposed to reflect different response patterns between respondents and nonrespondents. Data analysis and simulation studies show that Bayesian estimates based on the old three priors can be worse than the ML regardless of occurrence of boundary solution, contrary to previous studies. The Bayesian estimates from the two new priors are most preferable when a boundary solution occurs. We provide an illustrating example using data for a study of the relationship between a mother's smoking and her newborn's weight.     

Bayesian estimation of the generalized lognormal distribution using objective priors

The generalized lognormal distribution plays an important role in analysing data from different life testing experiments. In this paper, we consider Bayesian analysis of this distribution using various objective priors for the model parameters. Specifically, we derive expressions for the Jeffreys-type priors, the reference priors with different group orderings of the parameters, and the first-order matching priors. We also study the properties of the posterior distributions of the parameters under these improper priors. It is shown that only two of them result in proper posterior distributions. Numerical simulation studies are conducted to compare the performances of the Bayesian estimators under the considered priors and the maximum likelihood estimates. Finally, a real-data application is also provided for illustrative purposes.     

Bayesian Analysis of the Proportional Hazards Model with Time‐Varying Coefficients

We study a Bayesian analysis of the proportional hazards model with time‐varying coefficients. We consider two priors for time‐varying coefficients – one based on B‐spline basis functions and the other based on Gamma processes – and we use a beta process prior for the baseline hazard functions. We show that the two priors provide optimal posterior convergence rates (up to the term) and that the Bayes factor is consistent for testing the assumption of the proportional hazards when the two priors are used for an alternative hypothesis. In addition, adaptive priors are considered for theoretical investigation, in which the smoothness of the true function is assumed to be unknown, and prior distributions are assigned based on B‐splines.     

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.     

The use of Jeffreys priors for the Student-t distribution

The Jeffreys-rule prior and the marginal independence Jeffreys prior are recently proposed in Fonseca et al. [Objective Bayesian analysis for the Student-t regression model, Biometrika 95 (2008), pp. 325–333] as objective priors for the Student-t regression model. The authors showed that the priors provide proper posterior distributions and perform favourably in parameter estimation. Motivated by a practical financial risk management application, we compare the performance of the two Jeffreys priors with other priors proposed in the literature in a problem of estimating high quantiles for the Student-t model with unknown degrees of freedom. Through an asymptotic analysis and a simulation study, we show that both Jeffreys priors perform better in using a specific quantile of the Bayesian predictive distribution to approximate the true quantile.     

Bayesian sieve methods: approximation rates and adaptive posterior contraction rates

In the last 20 years, a lot of achievements have been made in the study of posterior contraction rates of nonparametric Bayesian methods, and plenty of them involve sieve priors, but mainly for specific models or sieves. We provide a posterior contraction theorem for general parametric sieve priors. The theorem has weaker and simpler conditions compared with the existing results, and indicates that the sieve prior is rate adaptive. We apply the general theorem to density estimations and nonparametric regression with jumps. We also provided a reversible jump MCMC (Markov Chain Monte Carlo) algorithm for the sieve prior.     

Default Bayesian analysis of the Behrens–Fisher problem

In the Bayesian approach, the Behrens–Fisher problem has been posed as one of estimation for the difference of two means. No Bayesian solution to the Behrens–Fisher testing problem has yet been given due, perhaps, to the fact that the conventional priors used are improper. While default Bayesian analysis can be carried out for estimation purposes, it poses difficulties for testing problems. This paper generates sensible intrinsic and fractional prior distributions for the Behrens–Fisher testing problem from the improper priors commonly used for estimation. It allows us to compute the Bayes factor to compare the null and the alternative hypotheses. This default procedure of model selection is compared with a frequentist test and the Bayesian information criterion. We find discrepancy in the sense that frequentist and Bayesian information criterion reject the null hypothesis for data, that the Bayes factor for intrinsic or fractional priors do not.     

Objective Bayesian hypothesis testing in regression models with first-order autoregressive residuals

This article considers the objective Bayesian testing in the normal regression models with first-order autoregressive residuals. We propose some solutions based on a Bayesian model selection procedure to this problem where no subjective input is considered. We construct the proper priors for testing the autocorrelation coefficient based on measures of divergence between competing models, which is called the divergence-based (DB) priors and then propose the objective Bayesian decision-theoretic rule, which is called the Bayesian reference criterion (BRC). Finally, we derive the intrinsic test statistic for testing the autocorrelation coefficient. The behavior of the Bayes factor-based DB priors is examined by comparing with the BRC in a simulation study and an example.     

Bayesian sample-size determination for one and two Poisson rate parameters with applications to quality control

We formulate Bayesian approaches to the problems of determining the required sample size for Bayesian interval estimators of a predetermined length for a single Poisson rate, for the difference between two Poisson rates, and for the ratio of two Poisson rates. We demonstrate the efficacy of our Bayesian-based sample-size determination method with two real-data quality-control examples and compare the results to frequentist sample-size determination methods.     

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.     

Constant hazard against a change-point alternative: a bayesian approach with censored data

A Bayesian approach to the problem of a constant hazard with a single change-point is developed using noninformative reference priors. We also present a generalization for the comparison for two treatments.     

Bayesian unit-root tests for Stochastic Volatility models

In this article, we consider Bayesian inference procedures to test for a unit root in Stochastic Volatility (SV) models. Unit-root tests for the persistence parameter of the SV models, based on the Bayes Factor (BF), have been recently introduced in the literature. In contrast, we propose a flexible class of priors that is non-informative over the entire support of the persistence parameter (including the non-stationarity region). In addition, we show that our model fitting procedure is computationally efficient (using the software WinBUGS). Finally, we show that our proposed test procedures have good frequentist properties in terms of achieving high statistical power, while maintaining low total error rates. We illustrate the above features of our method by extensive simulation studies, followed by an application to a real data set on exchange rates.     

Bayesian expectile regression with asymmetric normal distribution

In this paper, we adopt the Bayesian approach to expectile regression employing a likelihood function that is based on an asymmetric normal distribution. We demonstrate that improper uniform priors for the unknown model parameters yield a proper joint posterior. Three simulated data sets were generated to evaluate the proposed method which show that Bayesian expectile regression performs well and has different characteristics comparing with Bayesian quantile regression. We also apply this approach into two real data analysis.     

Bayesian shrinkage estimates for regression coefficients in m populations

For a linear regression model over m populations with separate regression coefficients but a common error variance, a Bayesian model is employed to obtain regression coefficient estimates which are shrunk toward an overall value. The formulation uses Normal priors on the coefficients and diffuse priors on the grand mean vectors, the error variance, and the between-to-error variance ratios. The posterior density of the parameters which were given diffuse priors is obtained. From this the posterior means and variances of regression coefficients and the predictive mean and variance of a future observation are obtained directly by numerical integration in the balanced case, and with the aid of series expansions in the approximately balanced case. An example is presented and worked out for the case of one predictor variable. The method is an extension of Box & Tiao's Bayesian estimation of means in the balanced one-way random effects model.