On the choice of the prior distribution in hypergeometric sampling

Information in a statistical procedure arising from sources other than sampling is called prior information, and its incorporation into the procedure forms the basis of the Bayesian approach to statistics. Under hypergeometric sampling, methodology is developed which quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information embedded in any prior distribution. The most conservative prior distribution from a specified class (each member of which carries the available prior information) is that prior distribution within the class over which the likelihood function has the greatest average domination. Four different families of prior distributions are developed by considering a Bayesian approach to the formation of lots. The most conservative prior distribution from each of the four families of prior distributions is determined and compared for the situation when no prior information is available. The results of the comparison advocate the use of the Polya (beta-binomial) prior distribution in hypergeometric sampling.     

Minimax Estimation of the Bounded Parameter of Some Discrete Distributions Under LINEX Loss Function

For a class of discrete distributions, including Poisson(θ), Generalized Poisson(θ), Borel(m, θ), etc., we consider minimax estimation of the parameter θ under the assumption it lies in a bounded interval of the form [0, m] and a LINEX loss function. Explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior are given. Also for Bernoulli(θ)-distribution, which is not in the mentioned class of discrete distributions, we give conditions for which the Bayes estimator of θ ∈ [0, m], m < 1 with respect to a boundary supported prior is minimax under LINEX loss function. Numerical values are given for the largest values of m for which the corresponding Bayes estimators of θ are minimax.     

Prbposterior expected loss as a scoring rule for prior distributions

A scoring rule for evaluating the usefulness of an assessed prior distribution should reflect the purpose for which the distribution is to be used. In this paper we suppose that sample data is to become available and that the posterior distribution will be used to estimate some quantity under a quadratic loss function. The utility of a prior distribution is consequently determined by its preposterior expected quadratic loss. It is shown that this loss function has properties desirable in a scoring rule and formulae are derived for calculating the scores it gives in some common problems. Many scoring rules give a very poor score to any improper prior distribution but, in contrast, the scoring rule proposed here provides a meaningful measure for comparing the usefulness of assessed prior distributions and non-informative (improper) prior distributions. Results for making this comparison in various situations are also given.     

On the bias and variance of some proportion estimators

Providing certain parameters are known, almost any linear map from R^P to R¹ can be adjusted to yield a consistent and unbiased estimator in the context of estimating the mixing proportion θ on the basis of an unclassified sample of observations taken from a mixture of two p-dimensional distributions in proportions θ and 1-θ. Attention is focused on an estimator proposed recently, θ, which has minimum variance over all such linear maps. Unfortunately, the form of θ depends on the means of the component distributions and the covariance matrix of the mixture distribution. The effect of using appropriate sample estimates for these unknown parameters in forming θ is investigated by deriving the asymptotic mean and variance of the resulting estimator. The relative efficiency of this estimator under normality is derived. Also, a study is undertaken of the performance of a similar type of estimator appropriate in the context where an observed data vector is not an observation from either one or the other onent distributions, but is recorded as an integrated measurement over a surface area which is a mixture of two categories whose characteristics have different statistical distributions.The asymptotic bias in this case is compared with some available practical results.     

A bayesian analysis of imperfect inspection models

An examination of sampling inspection models is provided for the case where inspection is not perfect and classification errors can be made. The conjugate family of distributions is obtained under t h e assumption that defective items are generated according to a Bernoulli process. To simplify analysis, the use of a single beta prior distribution is considered. Relevant predictive distributions are obtained.     

Bayesian analysis of a multiple-recapture model

In the Bayesian analysis of a multiple-recapture census, different diffuse prior distributions can lead to markedly different inferences about the population size N. Through consideration of the Fisher information matrix it is shown that the number of captures in each sample typically provides little information about N. This suggests that if there is no prior information about capture probabilities, then knowledge of just the sample sizes and not the number of recaptures should leave the distribution of Nunchanged. A prior model that has this property is identified and the posterior distribution is examined. In particular, asymptotic estimates of the posterior mean and variance are derived. Differences between Bayesian and classical point and interval estimators are illustrated through examples.     

Bayesian predictive modeling for Inverse Gamma-Pareto composite distribution

Inverse Gamma-Pareto composite distribution is considered as a model for heavy tailed data. The maximum likelihood (ML), smoothed empirical percentile (SM), and Bayes estimators (informative and non-informative) for the parameter θ, which is the boundary point for the supports of the two distributions are derived. A Bayesian predictive density is derived via a gamma prior for θ and the density is used to estimate risk measures. Accuracy of estimators of θ and the risk measures are assessed via simulation studies. It is shown that the informative Bayes estimator is consistently more accurate than ML, Smoothed, and the non-informative Bayes estimators.     

Numerical Bayesian inference with arbitrary prior

The purpose of the paper, is to explain how recent advances in Markov Chain Monte Carlo integration can facilitate the routine Bayesian analysis of the linear model when the prior distribution is completely user dependent. The method is based on a Metropolis-Hastings algorithm with a Student-t source distribution that can generate posterior moments as well as marginal posterior densities for model parameters. The method is illustrated with numerical examples where the combination of prior and likelihood information leads to multimodal posteriors due to prior-likelihood conflicts, and to cases where prior information can be summarized by symmetric stable Paretian distributions.     

ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE

In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ₁ ’ or ‘θ < θ₁ ’, where θ₁ is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ₁ is to replace an estimator, , of θ by the maximum of and θ₁. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ₁ , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ₁ , strictly exceed θ₁ except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ₁, and more generally produces estimators that respect the constraint in a smooth, realistic fashion.     

Model Selection Using Conditional Densities

In this article we propose a new method to select a discrete model f(x; θ), based on the conditional density of a sample given the value of a sufficient statistic for θ. The main idea is to work with a broad family of discrete distributions, called the family of power series distribution, for which there is a common sufficient statistic for the parameter of interest. The proposed method uses the maximum conditional density in order to select the best model.

We compare our proposal with the usual methodology based on Bayes factors. We provide several examples that show that our proposal works fine in most instances. Bayes factors are strongly dependent on the prior information about the parameters. Since our method does not require the specification of a prior distribution, it provides a useful alternative to Bayes factors.     

On Bayesian estimators in multistage binomial designs

A new class of Bayesian estimators for a proportion in multistage binomial designs is considered. Priors belong to the beta-J distribution family, which is derived from the Fisher information associated with the design. The transposition of the beta parameters of the Haldane and the uniform priors in fixed binomial experiments into the beta-J distribution yields bias-corrected versions of these priors in multistage designs. We show that the estimator of the posterior mean based on the corrected Haldane prior and the estimator of the posterior mode based on the corrected uniform prior have good frequentist properties. An easy-to-use approximation of the estimator of the posterior mode is provided. The new Bayesian estimators are compared to Whitehead's and the uniformly minimum variance estimators through several multistage designs. Last, the bias of the estimator of the posterior mode is derived for a particular case.     

Inequality indices and the starshaped principle of transfers

The evaluation of income distributions is usually based on the Pigou-Dalton (PD) principle which says that a transfer from any people to people who have less decreases economic inequality, i.e., increases the social evaluation index. We introduce two weaker principles of transfers which refer to a parameter θ. With the new principles, only those PD transfers increase the social evaluation index which take from the class of incomes above θ and give to the class below θ. The relative positions of individuals remain unchanged, and either no individual may cross the line θ (principle of transfers about θ) or some may do who have been situated next to it (starshaped principle of transfers at θ). θ may be a given constant, a function of mean income, or a quantile of the income distribution. The classes of indices which are consistent with these transfers are completely characterized, and examples are given.     

Bayesian Robustness Modelling of Location and Scale Parameters

Abstract. The modelling process in Bayesian Statistics constitutes the fundamental stage of the analysis, since depending on the chosen probability laws the inferences may vary considerably. This is particularly true when conflicts arise between two or more sources of information. For instance, inference in the presence of an outlier (which conflicts with the information provided by the other observations) can be highly dependent on the assumed sampling distribution. When heavy‐tailed (e.g. t) distributions are used, outliers may be rejected whereas this kind of robust inference is not available when we use light‐tailed (e.g. normal) distributions. A long literature has established sufficient conditions on location‐parameter models to resolve conflict in various ways. In this work, we consider a location–scale parameter structure, which is more complex than the single parameter cases because conflicts can arise between three sources of information, namely the likelihood, the prior distribution for the location parameter and the prior for the scale parameter. We establish sufficient conditions on the distributions in a location–scale model to resolve conflicts in different ways as a single observation tends to infinity. In addition, for each case, we explicitly give the limiting posterior distributions as the conflict becomes more extreme.     

A prior for the variance in hierarchical models

The choice of prior distributions for the variances can be important and quite difficult in Bayesian hierarchical and variance component models. For situations where little prior information is available, a ‘nonin-formative’ type prior is usually chosen. ‘Noninformative’ priors have been discussed by many authors and used in many contexts. However, care must be taken using these prior distributions as many are improper and thus, can lead to improper posterior distributions. Additionally, in small samples, these priors can be ‘informative’. In this paper, we investigate a proper ‘vague’ prior, the uniform shrinkage prior (Strawder-man 1971; Christiansen & Morris 1997). We discuss its properties and show how posterior distributions for common hierarchical models using this prior lead to proper posterior distributions. We also illustrate the attractive frequentist properties of this prior for a normal hierarchical model including testing and estimation. To conclude, we generalize this prior to the multivariate situation of a covariance matrix.     

Bayesian beta nonlinear models with constrained parameters to describe ruminal degradation kinetics

The models used to describe the kinetics of ruminal degradation are usually nonlinear models where the dependent variable is the proportion of degraded food. The method of least squares is the standard approach used to estimate the unknown parameters but this method can lead to unacceptable predictions. To solve this issue, a beta nonlinear model and the Bayesian perspective is proposed in this article. The application of standard methodologies to obtain prior distributions, such as the Jeffreys prior or the reference priors, involves serious difficulties here because this model is a nonlinear non-normal regression model, and the constrained parameters appear in the log-likelihood function through the Gamma function. This paper proposes an objective method to obtain the prior distribution, which can be applied to other models with similar complexity, can be easily implemented in OpenBUGS, and solves the problem of unacceptable predictions. The model is generalized to a larger class of models. The methodology was applied to real data with three models that were compared using the Deviance Information Criterion and the root mean square prediction error. A simulation study was performed to evaluate the coverage of the credible intervals.     

Sensitivity of a bayesian inference to prior assumptions

The sensitivity of-a Bayesian inference to prior assumptions is examined by Monte Carlo simulation for the beta-binomial conjugate family of distributions. Results for the effect on a Bayesian probability interval of the binomial parameter indicate that the Bayesian inference is for the most part quite sensitive to misspecification of the prior distribution. The magnitude of the sensitivity depends primarily on the difference of assigned means and variances from the respective means and variances of the actually-sampled prior distributions. The effect of a disparity in form between the assigned prior and actually-sampled distributions was less important for the cases tested.     

Empirical bayes estimates using the nonparametric maximum likelihood estimate for the priort†

This paper presents a smooth empirical Bayes estimation technique based on nonparametric maximum likelihood estimation of the prior distribution Posterior means based on this estimate of the prior are shown to be easily calculated for a variety of sampling situations Examples involving normal and binomial sampling are given.     

Failure probability estimation under additional subsystem information with application to semiconductor burn-in

In the classical approach to qualitative reliability demonstration, system failure probabilities are estimated based on a binomial sample drawn from the running production. In this paper, we show how to take account of additional available sampling information for some or even all subsystems of a current system under test with serial reliability structure. In that connection, we present two approaches, a frequentist and a Bayesian one, for assessing an upper bound for the failure probability of serial systems under binomial subsystem data. In the frequentist approach, we introduce (i) a new way of deriving the probability distribution for the number of system failures, which might be randomly assembled from the failed subsystems and (ii) a more accurate estimator for the Clopper–Pearson upper bound using a beta mixture distribution. In the Bayesian approach, however, we infer the posterior distribution for the system failure probability on the basis of the system/subsystem testing results and a prior distribution for the subsystem failure probabilities. We propose three different prior distributions and compare their performances in the context of high reliability testing. Finally, we apply the proposed methods to reduce the efforts of semiconductor burn-in studies by considering synergies such as comparable chip layers, among different chip technologies.     

A note on a modified double stage shrinkage estimator

A modified double stage shrinkage estimator has been proposed for the single parameter θ of a distribution function _Fθ. It is shown to be locally better in comparison to the usual double stage shrinkage estimator in the sense of smaller mean squared error in a certain neighbourhood of prior estimate θ_o of θ.     

Non-parametric estimation of an acceleration parameter

Let X₁,… X_m be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y₁,…,Y_n a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]^θ with θ being the unknown parameter under study.Define:U_ij=1 otherwise.The joint distribution of _ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the U_ii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ.