Probability Matching Priors Research Articles

A comparison of objective priors for Cronbach’s coefficient alpha using a balanced random effects model

. In this article, the reference and probability matching priors for Cronbach’s alpha will be derived. The performance of these two priors will be compared to that of the well-known Jeffreys prior and a divergence prior. Cronbach’s alpha is a measure used to assess the reliability of a set of test items. A simulation study will be considered to compare the performance of the priors, where the coverage rates, average interval lengths, and standard deviations of the interval lengths will be computed. A second simulation study will be considered where the mean relative error will be compared for the various priors using three different loss functions. The following loss functions will be considered, Squared error loss, Absolute error loss, and Linex loss. An illustrative example will also be considered. Throughout the article, the random effects approach will be used.

Bayesian analysis for the Shannon entropy of the Lomax distribution using noninformative priors

The Lomax distribution is one of the well-known distributions that is used to fit heavy-tailed data. In this paper, we investigate the estimation of Shannon entropy of the Lomax distribution using noninformative priors. Some important priors including Jeffreys prior, reference priors and probability matching priors are presented. We demonstrate that the reference and matching priors for the Lomax entropy do not match the reference and probability matching priors of the Lomax parameters, regardless of which parameter of interest is considered. The propriety and the existence of the expectation of the posterior under each prior are validated, respectively. A simulation study is conducted to assess the frequentist performance of the proposed Bayesian estimates in terms of the mean squared error and coverage probability. Finally, the approach is applied to three real data sets for illustrative purposes.

Hellinger Information Matrix and Hellinger Priors

Hellinger information as a local characteristic of parametric distribution families was first introduced in 2011. It is related to the much older concept of the Hellinger distance between two points in a parametric set. Under certain regularity conditions, the local behavior of the Hellinger distance is closely connected to Fisher information and the geometry of Riemann manifolds. Nonregular distributions (non-differentiable distribution densities, undefined Fisher information or denisities with support depending on the parameter), including uniform, require using analogues or extensions of Fisher information. Hellinger information may serve to construct information inequalities of the Cramer-Rao type, extending the lower bounds of the Bayes risk to the nonregular case. A construction of non-informative priors based on Hellinger information was also suggested by the author in 2011. Hellinger priors extend the Jeffreys rule to nonregular cases. For many examples, they are identical or close to the reference priors or probability matching priors. Most of the paper was dedicated to the one-dimensional case, but the matrix definition of Hellinger information was also introduced for higher dimensions. Conditions of existence and the nonnegative definite property of Hellinger information matrix were not discussed. Hellinger information for the vector parameter was applied by Yin et al. to problems of optimal experimental design. A special class of parametric problems was considered, requiring the directional definition of Hellinger information, but not a full construction of Hellinger information matrix. In the present paper, a general definition, the existence and nonnegative definite property of Hellinger information matrix is considered for nonregular settings.

Objective Bayesian analysis of the ratio of bivariate lognormal means

In this study, we consider the objective Bayesian analysis for the ratio of means in the bivariate lognormal distribution, in which properly assigning priors for the ratio of means is challenging due to the presence of nuisance parameters. As a result, we create probability matching priors and reference priors for the ratio of means. Jeffreys’ prior and the two group reference prior do not satisfy a first-order matching criterion. We also investigate the posterior distribution property for the general prior class, which includes Jeffreys’ prior, the reference prior, and the matching prior. Through a simulation study, we check the frequentist coverage probabilities and compare them with the generalized confidence interval method. It demonstrated that the proposed probability matching priors match very well with the target coverage probabilities even when the sample sizes are small in a frequentist sense. Two real examples are also provided.

Bayesian analysis for the Lomax model using noninformative priors

The Lomax distribution is an important member in the distribution family. In this paper, we systematically develop an objective Bayesian analysis of data from a Lomax distribution. Noninformative priors, including probability matching priors, the maximal data information (MDI) prior, Jeffreys prior and reference priors, are derived. The propriety of the posterior under each prior is subsequently validated. It is revealed that the MDI prior and one of the reference priors yield improper posteriors, and the other reference prior is a second-order probability matching prior. A simulation study is conducted to assess the frequentist performance of the proposed Bayesian approach. Finally, this approach along with the bootstrap method is applied to a real data set.

Application of objective priors for the multivariate Lomax distribution

For a model incorporating the effect of a common environment on several components of a system, a multivariate Lomax distribution (MLD) is generally considered by mixing exponential variables. Objective Bayesian has very good frequentist properties and provides a moderate solution for the prior elicitation which is one of important and difficult issues on Bayesian analysis. In this paper, we develop noninformative priors, such as the probability matching priors and reference priors, for the parameters of the MLD. We proved that a reference prior for the shape parameter is a first-order probability matching prior, but the reference priors for the scale parameters do not satisfy the first-order matching criterion. In addition, a second-order probability matching prior does not exist for all parameters. We also presented the conditions that make the posterior distributions for the general prior, including the probability matching prior and reference priors, to be proper. In particular, Jeffreys’ prior and probability matching priors for all parameters give proper posteriors, whereas reference priors for scale parameters give improper posteriors.

A simulation study to compare reference and other priors in the case of a standard univariate Student t-distribution

In this paper, reference and probability-matching priors are derived for the univariate Student t-distribution. These priors generally lead to procedures with properties frequentists can relate to while still retaining Bayes validity. The priors are tested by performing simulation studies. The focus is on the relative mean squared error from the posterior median and on the frequentist coverage of the 95% credibility intervals for a sample of size n = 30. Average interval lengths of the credibility intervals as well as the modes of the interval lengths based on 2 000 simulations are also considered. The performance of the priors is also tested on real data, namely daily logarithmic returns of IBM stocks. Keywords: Coverage percentages, Credibility intervals, Log-returns, Mean squared error, Probability-matching prior, Reference priors

Bayesian control charts for tolerance limits in the case of normal populations

By using air-lead data analysed by Krishnamoorthy and Mathew (2009) a Bayesian procedure is applied to obtain control limits for the upper one-sided tolerance limit. Reference and probability matching priors are derived for the pth quantile of a normal distribution. By simulating the predictive density of a future upper one-sided tolerance limit, "run-lengths" and average "run-lengths" are derived. In the second part of this paper control limits are derived for one-sided tolerance limits for the distribution of the difference between two normal random variables. This article illustrates the flexibility and unique features of the Bayesian simulation method for obtaining the posterior predictive distribution of a future one-sided tolerance limit. Keywords: Control charts, Probability-matching prior, Reference prior, Tolerance limits

Predictive probability matching priors for a certain non-regular model

Probability matching priors for Bayesian prediction in non-regular case are considered. For one-parameter family of distributions, the resulting priors match the posterior predictive quantile with the frequentist one up to the order of o(n−2), and they are solutions of a certain differential equation (denoted by matching equation). Although predictive probability matching priors depend on a nominal rate α in general, we provide a prior which satisfy the matching equation for every nominal rate α in non-regular location and scale models. A multi-parameter extension including location-scale model is also discussed.

Objective Bayesian analysis for generalized exponential stress–strength model

In reliability studies, a stress–strength model is often used to analyze a system that fails whenever the applied stress is greater than the strength. Statistical inference of reliability is widely used in a number of areas, such as engineering, clinical trials, and quality control. In addition to the common stress–strength model with one stress and one strength, the reliability of more complex systems has also been studied. In this study, we consider the reliability of a generalized stress–strength model that consists of a serial system with one stress and multiple strengths. We then develop the probability matching priors and reference priors for a generalized exponential stress–strength model. We demonstrate that the two-group reference prior and Jeffreys prior are not a matching prior. Through a simulation study and real data example, we also demonstrate that the proposed probability matching priors match the target coverage probabilities in a frequentist sense even for a small sample size.

Objective bayesian inference for quantile ratios in normal models

In medical research, it is important to compare quantiles of certain measures obtained from treatment and control groups, with the quantile ratio showing the effect of the treatment. In particular, inference of the quantile ratio based on large sample methods can be studied using a normal model. In this paper, we develop noninformative priors such as probability matching priors and reference priors for quantile ratios in normal models. It has been proved that the one-at-a-time reference prior satisfies a first-order matching criterion, while the Jeffreys’ and two-group reference priors do not when the variances are equal. Through simulation study and an example based on real data, we also confirm that the proposed probability matching priors match the target coverage probabilities in a frequentist sense even when the sample size is small.

Posterior propriety of bivariate lomax distribution under objective priors

The Lomax or Pareto II, distribution has been quite widely used for reliability modeling and life testing, and applied to the sizes of computer files on servers, and even application in the biological sciences. Especially, the bivariate Lomax distribution is considered for a two components system which works under interdependency circumstances. We here develop the noninformative priors such as the probability matching priors and the reference priors for the parameters in the bivariate lomax population. It turns out that the reference prior satisfy a first order matching criterion only for some parameter groupings. We also check the conditions for the propriety of posterior distributions under the general prior class including the matching priors and the reference priors. It is a quite interesting that the posterior distributions under the reference priors for some parameter groupings are proper, but those for the other parameter groupings are improper. However all of the matching priors give the proper posteriors.

Non-subjective priors for wrapped Cauchy distributions

Circular data can arise from many sources, such as image processing, protein structure, and geological data, just to name a few. Wrapped stable family of distributions constitute one of the most widely used class of distributions for the analysis of such data. Wrapped Cauchy distribution is a member of this family and it is the only one known to have a single term explicit pdf compared to the infinite series representations for all the others. We develop in this paper reference priors and probability matching priors for this distribution, which turn out to be quite elegant and provide good frequentist performance as well.

Probability matching priors for the powers of mean and variance in the normal distribution

Probability matching priors for the powers of mean and variance in the normal distribution

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

ABSTRACTIn 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 inference on extreme value distribution using upper record values

ABSTRACTIn this paper we address estimation and prediction problems for extreme value distributions under the assumption that the only available data are the record values. We provide some properties and pivotal quantities, and derive unbiased estimators for the location and rate parameters based on these properties and pivotal quantities. In addition, we discuss mean-squared errors of the proposed estimators and exact confidence intervals for the rate parameter. In Bayesian inference, we develop objective Bayesian analysis by deriving non informative priors such as the Jeffrey, reference, and probability matching priors for the location and rate parameters. We examine the validity of the proposed methods through Monte Carlo simulations for various record values of size and present a real data set for illustration purposes.

Objective Bayes models for compatibility assessment and bias estimation

The paper derives a class of non-informative, probability matching priors and of default, data-dependent priors for the difference in two normal means when the variance(s) are unknown. These distributions are useful to get a Bayesian model in the situation where one has to assess compatibility of data obtained by a laboratory with the specified values and uncertainty estimate given in the certificate of analysis. The latter is then interpreted as the parameter assignment for measurand’s normal distribution. The Bayes rules arising from these prior distributions are compared via the frequentist coverage probability of corresponding credible intervals for lab’s bias.

A Bayesian control chart for a common coefficient of variation

ABSTRACTBy using the medical data analyzed by Kang et al. (2007), a Bayesian procedure is applied to obtain control limits for the coefficient of variation. Reference and probability matching priors are derived for a common coefficient of variation across the range of sample values. By simulating the posterior predictive density function of a future coefficient of variation, it is shown that the control limits are effectively identical to those obtained by Kang et al. (2007) for the specific dataset they used. This article illustrates the flexibility and unique features of the Bayesian simulation method for obtaining posterior distributions, predictive intervals, and run-lengths in the case of the coefficient of variation. A simulation study shows that the 95% Bayesian confidence intervals for the coefficient of variation have the correct frequentist coverage.

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.

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.