Posterior Analysis for Normalized Random Measures with Independent Increments

Abstract.  One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. In this paper, we provide a comprehensive Bayesian non-parametric analysis of random probabilities which are obtained by normalizing random measures with independent increments (NRMI). Special cases of these priors have already shown to be useful for statistical applications such as mixture models and species sampling problems. However, in order to fully exploit these priors, the derivation of the posterior distribution of NRMIs is crucial: here we achieve this goal and, indeed, provide explicit and tractable expressions suitable for practical implementation. The posterior distribution of an NRMI turns out to be a mixture with respect to the distribution of a specific latent variable. The analysis is completed by the derivation of the corresponding predictive distributions and by a thorough investigation of the marginal structure. These results allow to derive a generalized Blackwell–MacQueen sampling scheme, which is then adapted to cover also mixture models driven by general NRMIs.     

Nonparametric density deconvolution by weighted kernel estimators

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of negative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.     

ASSOCIATIONS BETWEEN AIR POLLUTION AND HOSPITAL VISITS FOR CARDIOVASCULAR DISEASES IN THE ELDERLY IN SYDNEY USING BAYESIAN STATISTICAL METHODS

Using generalized linear models (GLMs), Jalaludin  et al. (2006;  J. Exposure Analysis and Epidemiology   16 , 225–237) studied the association between the daily number of visits to emergency departments for cardiovascular disease by the elderly (65+) and five measures of ambient air pollution. Bayesian methods provide an alternative approach to classical time series modelling and are starting to be more widely used. This paper considers Bayesian methods using the dataset used by Jalaludin  et al.  (2006) , and compares the results from Bayesian methods with those obtained by Jalaludin  et al.  (2006) using GLM methods.     

Adjustment for nonresponse with variables from different sources: Bias correction and precision loss,with application to the Swiss European Social Survey 2012

ABSTRACT

Adjustment for nonresponse should reduce the nonresponse bias without decreasing the precision of the estimates. Adjustment for nonresponses are commonly based on socio-demographic variables, although these variables may be poorly correlated with response propensities and with variables of interest. Such variables nevertheless have the advantage of being available for all sample units, whether or not they are participating in the survey. Alternatively, adjustment for nonresponse can be obtained from a follow-up survey aimed at sample units which did not participate in the survey and from which the variables are designed to be correlated with response propensities. However, information collected through these follow-up surveys is not available for people in the sample who participated neither in the survey nor in its nonresponse follow-up. These two sets of variables when used in a nonresponse model for the Swiss European Social Survey 2012 differ only slightly with regard to their effect on bias correction and on the precision of estimates. The variables from the follow-up are performing slightly better. In both cases, the adjustment for nonresponse performs poorly.     

Fast robust feature screening for ultrahigh-dimensional varying coefficient models

In this paper, we propose a new partial correlation, the so-called composite quantile partial correlation, to measure the relationship of two variables given other variables. We further use this correlation to screen variables in ultrahigh-dimensional varying coefficient models. Our proposed method is fast and robust against outliers and can be efficiently employed in both single index variable and multiple index variable varying coefficient models. Numerical results indicate the preference of our proposed method.     

A backward construction and simulation of correlated Poisson processes

In this paper, we consider a generalisation of the backward simulation method of Duch et al. [New approaches to operational risk modeling. IBM J Res Develop. 2014;58:1–9] to build bivariate Poisson processes with flexible time correlation structures, and to simulate the arrival times of the processes. The proposed backward construction approach uses the Marshall–Olkin bivariate binomial distribution for the conditional law and some well-known families of bivariate copulas for the joint success probability in lieu of the typical conditional independence assumption. The resulting bivariate Poisson process can exhibit various time correlation structures which are commonly observed in real data.     

Empirical Uncertain Bayes Methods in Area‐level Models

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area‐level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus, it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one‐point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area‐level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson‐gamma and the binomial‐beta, and the normal‐normal (Fay–Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well‐known expectation‐maximization algorithm and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second‐order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.     

Tests for Structural Changes in Time Series of Counts

We propose methods for detecting structural changes in time series with discrete‐valued observations. The detector statistics come in familiar L2‐type formulations incorporating the empirical probability generating function. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. For both models, we study mainly structural changes due to a change in distribution, but we also comment for the classical problem of parameter change. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is also included along with a real data example.     

Score test for homogeneity of dispersion in generalized Poisson mixed models with excess zeros

In many applications, the clustered count data often contain excess zeros and the zero-inflated generalized Poisson mixed (ZIGPM) regression model may be suitable. However, dispersion in ZIGPM is often treated as fixed unknown parameter, and this assumption may be not appropriate in some situations. In this article, a score test for homogeneity of dispersion parameter in ZIGPM regression model is developed and corresponding test statistic is obtained. Sampling distribution and power of the score test statistic are investigated through Monte Carlo simulation. Finally, results from a biological example illustrate the usefulness of the diagnostic statistic.