Reference Priors Research Articles

Default Bayesian testing for scale parameters in the log-logistic distributions

This paper deals with the problem of testing on the equality of the scale parameters in the log-logistic distributions. We propose default Bayesian testing procedures for the scale parameters under the reference priors. The reference prior is usually improper which yields a calibration problem that makes the Bayes factor to be dened up to a multiplicative constant. Therefore, we propose the default Bayesian testing procedures based on the fractional Bayes factor and the intrinsic Bayes factor under the reference priors. To justify proposed procedures, a simulation study is provided and also, an example is given.

Default Bayesian one sided testing for the shape parameter in the log-logistic distribution

This paper deals with the problem of testing on the shape parameter in the log-logistic distribution. We propose default Bayesian testing procedures for the shape parameter under the reference priors. The reference prior is usually improper which yields a calibration problem that makes the Bayes factor to be defined up to a multiplicative constant. We can solve the this problem by the intrinsic Bayes factor and the fractional Bayes factor. Therefore we propose the default Bayesian testing procedures based on the fractional Bayes factor and the intrinsic Bayes factors under the reference priors. Simulation study and an example are provided.

Do single-arm trials have a role in drug development plans incorporating randomised trials?

Often, single‐arm trials are used in phase II to gather the first evidence of an oncological drug's efficacy, with drug activity determined through tumour response using the RECIST criterion. Provided the null hypothesis of ‘insufficient drug activity’ is rejected, the next step could be a randomised two‐arm trial. However, single‐arm trials may provide a biased treatment effect because of patient selection, and thus, this development plan may not be an efficient use of resources. Therefore, we compare the performance of development plans consisting of single‐arm trials followed by randomised two‐arm trials with stand‐alone single‐stage or group sequential randomised two‐arm trials. Through this, we are able to investigate the utility of single‐arm trials and determine the most efficient drug development plans, setting our work in the context of a published single‐arm non‐small‐cell lung cancer trial. Reference priors, reflecting the opinions of ‘sceptical’ and ‘enthusiastic’ investigators, are used to quantify and guide the suitability of single‐arm trials in this setting. We observe that the explored development plans incorporating single‐arm trials are often non‐optimal. Moreover, even the most pessimistic reference priors have a considerable probability in favour of alternative plans. Analysis suggests expected sample size savings of up to 25% could have been made, and the issues associated with single‐arm trials avoided, for the non‐small‐cell lung cancer treatment through direct progression to a group sequential randomised two‐arm trial. Careful consideration should thus be given to the use of single‐arm trials in oncological drug development when a randomised trial will follow. Copyright © 2015 The Authors. Pharmaceutical Statistics published by JohnWiley & Sons Ltd.

Bayesian analysis of two-piece location–scale models under reference priors with partial information

Bayesian estimators are developed and compared with the maximum likelihood estimators for the two-piece location–scale models, which contain several well-known distributions such as the asymmetric Laplace distribution, the two-piece normal distribution, and the two-piece Student-t distribution. For the validity of Bayesian analysis, it is essential to use priors that could lead to proper posterior distributions. Specifically, reference priors with partial information have been considered. A sufficient and necessary condition is established to guarantee the propriety of the posterior distribution under a general class of priors. The performance of the proposed approach is illustrated through extensive simulation studies and real data analysis.

Noninformative priors for the ratio of the shape parameters of two Weibull distributions

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. Here, the noninformative priors for the ratio of the shape parameters of two Weibull models are introduced. The first criterion used is the asymptotic matching of the coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. We develop the probability matching priors for the ratio of the shape parameters using the following matching criteria: quantile matching, matching of the distribution function, highest posterior density matching, and matching via inversion of the test statistics. We obtain one particular prior that meets all the matching criteria. Next, we derive the reference priors for different groups of ordering. Our findings show that some of the reference priors satisfy a first-order matching criterion and the one-at-a-time reference prior is a second-order matching prior. Lastly, we perform a simulation study and provide a real-world example.

Objective Bayesian analysis accelerated degradation test based on Wiener process models

Constant-stress accelerated degradation test (CSADT) as an effective model is widely used in assessing product reliability when measurements of degradation leading to failure can be observed. We model the degradation process as a Wiener process. In this paper, we make inferences about the parameters of the CSADT using an objective Bayesian method. The noninformative priors (Jefferys prior and two reference priors) are derived, and we show that their posterior distributions are proper. Since the posterior distributions are very complicated, Gibbs sampling algorithms for the Bayesian inference based on the Jefferys prior and two reference priors are proposed. Some simulation studies are conducted to show the effectiveness of the objective Bayesian analysis. Finally, we apply the objective Bayesian method to a real data set and estimate the mean-time-to-failure (MTTF) under use condition.

Noninformative priors for the common shape parameters of Weibull distributions

The Weibull distribution is widely used in survival analysis, reliability engineering, and failure analysis. In this paper, we develop noninformative priors such as probability matching priors for the common shape parameter of Weibull models using the asymptotic matching of coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. Here, various matching criteria, such as quantile matching, matching of distribution function, highest posterior density matching, and matching via inversion of test statistics, are considered. Next, we derive the reference priors for groups of ordering. It also shows that most reference priors do not satisfy the first order matching criterion, but the one-at-a-time reference prior is the second order matching priors. A simulation study is performed and a real example is given.

Analysis of Frechet Distribution Using Reference Priors

This article develops the Bayesian estimators in the context of reference priors for the two-parameter Frechet distribution. The general forms of the second-order matching priors are also derived in case of any parameter of interest and concluded that the reference prior is also a second order matching prior. Since the Bayesian estimators cannot be obtained in closed form, they are obtained using Monte Carlo simulation and Laplace approximation. The Bayesian and maximum likelihood estimates are compared via simulation study. Two real-life data sets are analyzed for illustration and comparison purpose.

Overall Objective Priors

In multi-parameter models, reference priors typically depend on the parameter or quantity of interest, and it is well known that this is necessary to produce objective posterior distributions with optimal properties. There are, however, many situations where one is simultaneously interested in all the parameters of the model or, more realistically, in functions of them that include aspects such as prediction, and it would then be useful to have a single objective prior that could safely be used to produce reasonable posterior inferences for all the quantities of interest. In this paper, we consider three methods for selecting a single objective prior and study, in a variety of problems including the multinomial problem, whether or not the resulting prior is a reasonable overall prior.

Bayesian analysis of constant-stress accelerated life test for the Weibull distribution using noninformative priors

Noninformative priors are used in estimating the model parameters in constant-stress accelerated life tests (ALTs) under Type-II censoring when the Weibull distributions are assumed. Several reference priors and the general forms of the probability matching priors are derived. It is verified that one of the reference priors is the unique second-order jointly probability matching prior. The propriety of the posteriors based on the reference priors is shown. It shows that the matching prior performs better than the other two reference priors in meeting the target coverage probabilities. A real data set from Nelson (1990) is analyzed for illustration.

Noninformative priors for the common shape parameter of several inverse Gaussian distributions

Abstract In this paper, we develop the noninformative priors for the common shape parameterof several inverse Gaussian distributions. Specially, we want to develop noninformativepriors which satisfy certain objective criterion. The probability matching priors andreference priors of the common shape parameter will be developed. It turns out thatthe second order matching prior does not exist. The reference priors satisfy the rstorder matching criterion, but Je rey’s prior is not the rst order matching prior. Weshowed that the proposed reference prior matches the target coverage probabilities in afrequentist sense through simulation study, and an example based on real data is given.Keywords: Inverse Gaussian distribution, matching prior, reference prior, shape param-eter. 1. Introduction The probability density function (pdf) of inverse gaussian with parameters and isgiven byf(x) =r2ˇx 3 2 expˆ(x ) 2 2 2 x˙;x>0; (1.1)where >0 is mean parameter and >0 is the scale parameter. The shape parameter forthe inverse Gaussian distribution is de ned as , where = =. Because of the versatilityand exibility in modelling right-skewed data, the inverse gaussian distribution has poten-tially useful applications in a wide variety of elds such as biology, economics, reliabilitytheory and life testing as discussed in Chhikara and Folks (1989) and Seshadri (1999).The present paper focuses on noninformative priors for the common shape parameter ofseveral inverse Gaussian distributions. The shape parameter of inverse Gaussian distributionhas an important meaning. For example, Chhikara and Folks (1989) derived an exact methodof seeking a con dence interval for the ratio of means when the shape parameters of these

Objective Bayesian hypothesis testing in binomial regression models with integral prior distributions

In this work we apply the methodology of integral priors to deal with Bayesian model selection in nested binomial regression models with a general link function. These models are often used to investigate associations and risks in epidemiological studies where one goal is to find whether or not an exposure is a risk factor for developing a certain disease; the purpose of the current paper is to test the effect of specific exposure factors. We formulate the problem as a Bayesian model selection one and solve it using objective Bayes factors. To elicit prior distributions on the regression coefficients of the binomial regression models, we rely on the methodology of integral priors that is nearly automatic as it only requires the specification of estimation reference priors and it does not depend on tuning parameters or on hyperparameters.

Hellinger Distance and Non-informative Priors

This paper introduces an extension of the Jeffreys’ rule to the construction of objective priors for non-regular parametric families. A new class of priors based on Hellinger information is introduced as Hellinger priors. The main results establish the relationship of Hellinger priors to the Jeffreys’ rule priors in the regular case, and to the reference and probability matching priors for the non-regular class introduced by Ghosal and Samanta. These priors are also studied for some non-regular examples outside of this class. Their behavior proves to be similar to that of the reference priors considered by Berger, Bernardo, and Sun, however some differences are observed. For the multi-parameter case, a combination of Hellinger priors and reference priors is suggested and some examples are considered.

Default Bayesian testing for the equality of shape parameters in the inverse Weibull distributions

Abstract This article deals with the problem of testing for the equality of the shape pa-rameters in two inverse Weibull distributions. We propose Bayesian hypothesis testingprocedures for the equality of the shape parameters under the noninformative prior.The noninformative prior is usually improper which yields a calibration problem thatmakes the Bayes factor to be de ned up to a multiplicative constant. So we propose thedefault Bayesian hypothesis testing procedures based on the fractional Bayes factor andthe intrinsic Bayes factors under the reference priors. Simulation study and an exampleare provided.Keywords: Fractional Bayes factor, intrinsic Bayes factor, inverse Weibull distribution,reference prior, shape parameter. 1. Introduction Consider Xhas an inverse Weibull distribution with the scale parameter and the shapeparameter . Then the likelihood function isL(; ) =  x ( +1) expˆx˙;x>0; (1.1)where >0 and >0. Drapella (1993) calls the inverse Weibull distribution as the comple-mentary Weibull distribution and Mudhokar and Kollia (1994) call it the reciprocal Weibulldistribution. The inverse Weibull distribution has the ability to model failure rates whichare quite common in reliability (Keller and Kamath, 1982; Erto, 1989; Calabria and Pulcini,1989). The density and the hazard function of inverse Weibull can be unimodal or decreas-ing depending on the choice of the shape parameter. Also the inverse Weibull distributionbecomes the inverse Rayleigh distribution and the inverse exponential distribution for = 2and = 1, respectively.The present paper focuses on Bayesian testing for the equality of the shape parametersin two inverse Weibull distributions. In Bayesian model selection or testing problem, theBayes factor under proper priors or informative priors have been very successful. However,

Default Bayesian testing for the equality of the scale parameters of several inverted exponential distributions

This article deals with the problem of testing the equality of the scale parameters of several inverted exponential distributions. We propose Bayesian hypothesis testing procedures for the equality of the scale parameters under the noninformative prior. The noninformative prior is usually improper which yields a calibration problem that makes the Bayes factor to be defined up to a multiplicative constant. So we propose the default Bayesian hypothesis testing procedures based on the fractional Bayes factor and the intrinsic Bayes factors under the reference priors. Simulation study and an example are provided.

On Divergence Measures Leading to Jeffreys and Other Reference Priors

The paper presents new measures of divergence between prior and posterior which are maximized by the Jeffreys prior. We provide two methods for proving this, one of which provides an easy to verify sufficient condition. We use such divergences to measure information in a prior and also obtain new objective priors outside the class of Bernardo’s reference priors.

Objective Bayes analysis of zero-inflated Poisson distribution with application to healthcare data

In this article, non-informative priors are investigated for a zero-inflated Poisson distribution with two parameters: the probability of zeros and the mean of the Poisson part. Both the reference prior and the Jeffreys prior are derived and shown to be second-order matching priors when only the mean of the Poisson part is of interest. However, when the probability of zeros is of interest, the reference prior is still a second-order matching prior, whereas the Jeffreys prior is not so. Furthermore, when both parameters are of interest, the reference prior is a unique second-order matching prior. Frequentist coverage probabilities of the posterior confidence sets based on the Jeffreys and reference priors are compared with each other using Monte Carlo simulations and with confidence sets based on the maximum likelihood estimation.

Default Bayesian hypothesis testing for the scale parameters in the half logistic distributions

This article deals with the problem of testing the equality of the scale parameters in the half logistic distributions. We propose Bayesian hypothesis testing procedures for the equality of the scale parameters under the noninformative priors. The noninformative prior is usually improper which yields a calibration problem that makes the Bayes factor to be dened up to a multiplicative constant. Thus we propose the default Bayesian hypothesis testing procedures based on the fractional Bayes factor and the intrinsic Bayes factors under the reference priors. Simulation study and an example are provided.

Inferences on Lung Cancer Mortality Rates Based on Reference Priors Under Partial Ordering

Summary We present a novel analysis of a landmark table of dose–response mortality counts from lung cancer in men. The data were originally collected by Doll and Hill. Our inferences are based on Poisson models for which the rates of occurrence are partially ordered according to two covariates. The partial ordering of the mortality rates enforces the well-established knowledge that lung cancer mortality rates are higher for older men and for heavier smokers. The ordered group reference priors that we use in our analyses generalize a class of reference priors that we previously derived for models of count data in which the rates of occurrence in different categories are completely ordered with respect to the values of a single covariate. The reference models for the lung cancer data based on the proposed priors are more flexible than and can be superior, in terms of goodness of fit, to a Bayesian version of several parametric models derived from a mathematical theory of carcinogenesis that have appeared in the literature.

Objective Bayesian Analysis for Recurrent Events in Presence of Competing Risks

HeIn this article, we use Bayesian reference analysis to consider the model of recurrent events with competing risks, where the sampling distribution of the observations due to some failure types is modeled through a proportional intensity homogeneous Poisson process and individuals are expected to experience repeated events along with the fixed covariates. The goal is to estimate the cause-specific intensity functions. Three reference priors are derived according to this model. Their associated posteriors are also discussed. The simulation work and real data application based on the reference priors and the classical likelihood approach show the efficiency of reference analysis.