Bayesian prediction based on a class of shrinkage priors for location-scale models

A class of shrinkage priors for multivariate location-scale models is introduced. We consider Bayesian predictive densities for location-scale models and evaluate performance of them using the Kullback–Leibler divergence. We show that Bayesian predictive densities based on priors in the introduced class asymptotically dominate the best invariant predictive density.     

A note on shrinkage wavelet estimation in Bayesian analysis

In this short note the closed form of the soft wavelet shrinkage estimator is derived, extending the work of Huang (2002) for the scale mixture of normal distributions.     

Bayesian model averaging and model selection: two sides of the same coin when identifying the determinants of trade union density?

One of the main problems in empirical sciences is the uncertainty about the relevance of variables. In the debate on the variables that provide a systematic and robust explanation of the share of employees that are members of trade unions, i.e. of trade union density, the problem of variable uncertainty is striking. In regression analyses there is the problem of having to select variables. One problem in the union density discussion is that depending on the chosen combination of regressors different results in the identification of relevant variables are achieved. To systematically analyze which variables are relevant the literature suggests model averaging and selection strategies. While the two strategies have advantages and disadvantages, the aim of this paper is to apply both. Based on a characteristic cross-country panel data set we find differences and similarities based on our evaluation and ask whether a methodological triangulation is possible.     

A class of multiple shrinkage estimators

Based on a sample of size n, we investigate a class of estimators of the mean of a p-variate normal distribution with independent components having unknown covariance. This class includes the James-Stein estimator and Lindley's estimator as special cases and was proposed by Stein. The mean squares error improves on that of the sample mean for p3. Simple approximations imations for this improvement are given for large n or p. Lindley's estimator improves on that of James and Stein if either n is large, and the coefficient of variation of is less than a certain increasing function of p, or if p is large. An adaptive estimator is given which for large samples always performs at least as well as these two estimators.     

Efficient robust tests in parametric models

Summary Let {P :}, an open subset of R ^k , be a regular parametric model for a sample of n independent, identically distributed observations. Formulated and solved in this paper is a robust version of the classical multi-sided hypothesis testing problem concerning , or a subvector of . In the robust testing problem, the usual parametric null hypothesis and alternatives are both replaced with larger, more realistic, sets of possible distributions for each observation. These sets, defined in terms of a Hellinger metric projection of the actual distribution onto a subspace associated with the parametric null hypothesis, are required to shrink as sample size increases, so as to avoid trivial asymptotics. One construction of an asymptotically minimax test for the robust testing problem is based upon the robust estimate of developed in Beran (1979); another construction amounts to an adaptively modified C() test.Research supported by National Science Foundation Grant MCS 75-10376     

Efficient robust estimates in parametric models

Summary Let {P ⁿ :}, an open subset ofR ^k , be a regular parametric model for a sample ofn independent, identically distributed observations. This paper describes estimates {T _n ;n1} of which are asymptotically efficient under the parametric model and are robust under small deviations from that model. In essence, the estimates are adaptively modified, one-step maximum likelihood estimates, which adjust themselves according to how well the parametric model appears to fit the data. When the fit seems poor,T _n discounts observations that would have large influence on the value of the usual one-step MLE. The estimates {T _n } are shown to be asymptotically minimax, in the Hájek-LeCam sense, for a Hellinger ball contamination model. An alternative construction of robust asymptotically minimax estimates, as modified MLE's, is described for canonical exponential families.This research was supported in part by National Science Foundation Grant MCS 75-10376     

Uniformly robust tests in errors-in-variables models

Consider an ordinary errors-in-variables model. The true level α _n (θ*) of a test at nominal level α and sample size n is said to be pointwise robust if α _n (θ*) → α as n → ∞ for each parameter θ*. Let Ω* be a set of values of θ*. Define α _n = sup _{θ* ∈Ω*}α _n (θ*). The test is said to be uniformly robust over Ω* if α _n → α as n → ∞. Corresponding definitions apply to the coverage probabilities of confidence sets. It is known that all existing large-sample tests for the parameters of the errors-in-variables model are pointwise robust. However, they might not be uniformly robust over certain null parameter spaces. In this paper, we construct uniformly robust tests for testing the vector coefficient parameter and vector slope parameter in the functional errors-in-variables model. These tests are established through constructing the confidence sets for the same parameters in the model with similar desirable property. Power comparisons based on simulation studies between the proposed tests and some existing tests in finite samples are also presented.     

Stability and fairness in models with a multiple membership

This article studies a model of coalition formation for the joint production (and finance) of public projects, in which agents may belong to multiple coalitions. We show that, if projects are divisible, there always exists a stable (secession-proof) structure, i.e., a structure in which no coalition would reject a proposed arrangement. When projects are indivisible, stable allocations may fail to exist and, for those cases, we resort to the least core in order to estimate the degree of instability. We also examine the compatibility of stability and fairness in metric environments with indivisible projects, where we also explore the performance of well-known solutions, such as the Shapley value and the nucleolus.     

A review of Bayesian dynamic forecasting models: Applications in marketing

We briefly review the main developments of Bayesian dynamic models. The emphasis is on marketing applications. Typical examples in this area are discussed. The concepts of monitoring and intervention are carefully explained with illustrative examples and open source computational routines. We avoid algebraic developments and instead use graphical examples to illustrate theoretical aspects. Two real-world problems using Bayesian dynamic models are discussed. Finally, we describe recent developments and alternative proposals to formally address the dependence when dealing with the modeling of multiple time series.     

Schatten class membership of Hankel operators on the unit sphere

Let H²(S) be the Hardy space on the unit sphere S in Cn, n?2. Consider the Hankel operator Hf=(1−P)Mf|H²(S), where the symbol function f is allowed to be arbitrary in L²(S,dσ). We show that for p>2n, Hf is in the Schatten class Cp if and only if f−Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if” part. We also show that the membership Hf∈C2n implies f−Pf=0, i.e., Hf=0.     

Bayesian selection of graphical regulatory models

We define a new class of coloured graphical models, called regulatory graphs. These graphs have their own distinctive formal semantics and can directly represent typical qualitative hypotheses about regulatory processes like those described by various biological mechanisms. They admit an embellishment into classes of probabilistic statistical models and so standard Bayesian methods of model selection can be used to choose promising candidate explanations of regulation. Regulation is modelled by the existence of a deterministic relationship between the longitudinal series of observations labelled by the receiving vertex and the donating one. This class contains longitudinal cluster models as a degenerate graph. Edge colours directly distinguish important features of the mechanism like inhibition and excitation and graphs are often cyclic. With appropriate distributional assumptions, because the regulatory relationships map onto each other through a group structure, it is possible to define a conditional conjugate analysis. This means that even when the model space is huge it is nevertheless feasible, using a Bayesian MAP search, to a discover regulatory network with a high Bayes Factor score. We also show that, like the class of Bayesian Networks, regulatory graphs also admit a formal but distinctive causal algebra. The topology of the graph then represents collections of hypotheses about the predicted effect of controlling the process by tearing out message passers or forcing them to transmit certain signals. We illustrate our methods on a microarray experiment measuring the expression of thousands of genes as a longitudinal series where the scientific interest lies in the circadian regulation of these plants.     

Bayesian analysis of fixed-interval preventive-maintenance models

This article develops methods for making accurate decisionswhen scheduling preventive maintenance in systems where inter-eventtimes can be modelled by a delayed renewal process or delayedalternating renewal process. A practical application, relatingto the reliability and maintenance of a relatively low-levelcomponent (valve) in a continuous-process industry over a periodof six years is presented to demonstrate and compare the differentapproaches. Our analyses indicate a cost-effective recommendationfor maintenance practice in this context. Our main thrust relates to the use of Bayesian methodology inorder to obtain rational, admissible decisions. Particular advancesover previous research allow for informative prior distributions,better approximations which lead to improved accuracy, non-negligibledowntimes, and general lifetime distributions. General analyticsolutions are sought for the simpler models, in order to achieveaccuracy and insight. The resulting integrals can only be solvedto give an infinite series and one approximation to the requiredsolution is obtained by truncating this series. Two other approximationsare developed, based on expansions of the prior predictive andlog-posterior distributions. A simulation approach is also developed to include prior informationand hence provide alternative approximations to these optimaldecisions. With exponential lifetime distributions, the relevantposterior lifetime distributions are non-central Pareto. Thissimulation is simple to program, compared to the approximations,but requires more computing time. It is accurate and extendseasily to situations involving greater complexity. We considertwo such extensions, the inclusion of downtimes and Weibulllifetime distributions.     

Bayesian analysis of conditional autoregressive models

Conditional autoregressive (CAR) models have been extensively used for the analysis of spatial data in diverse areas, such as demography, economy, epidemiology and geography, as models for both latent and observed variables. In the latter case, the most common inferential method has been maximum likelihood, and the Bayesian approach has not been used much. This work proposes default (automatic) Bayesian analyses of CAR models. Two versions of Jeffreys prior, the independence Jeffreys and Jeffreys-rule priors, are derived for the parameters of CAR models and properties of the priors and resulting posterior distributions are obtained. The two priors and their respective posteriors are compared based on simulated data. Also, frequentist properties of inferences based on maximum likelihood are compared with those based on the Jeffreys priors and the uniform prior. Finally, the proposed Bayesian analysis is illustrated by fitting a CAR model to a phosphate dataset from an archaeological region.     

Evaluation of fuzzy linear regression models by comparing membership functions

Fuzzy linear regression models can provide an estimated fuzzy number that has a fuzzy membership function. If a point that has the highest membership value from the estimated fuzzy number is not within the support of the observed fuzzy membership function, a decision-maker can have high risk from the estimate. In this study a modification of fuzzy linear regression analysis based on a criterion of minimizing the difference of the fuzzy membership values between the observed and estimated fuzzy numbers is proposed. Two numerical examples are used to evaluate the fuzzy regression models.     

Heteroscedastic normal-exponential mixture models: Bayesian and classical approaches

In this paper, we introduce a Bayesian analysis for mixture of distributions belonging to the exponential family. As a special case we consider a mixture of normal exponential distributions including joint modeling of the mean and variance. We also consider joint modeling of the mean and variance heterogeneity. Markov Chain Monte Carlo (MCMC) methods are used to obtain the posterior summaries of interest. We also introduce and apply an EM algorithm, where the maximization is obtained applying the Fisher scoring algorithm. Finally, we also include analysis of real data sets to illustrate the proposed methodology.