On the shrinkage of local linear curve estimators

Local linear curve estimators are typically constructed using a compactly supported kernel, which minimizes edge effects and (in the case of the Epanechnikov kernel) optimizes asymptotic performance in a mean square sense. The use of compactly supported kernels can produce numerical problems, however. A common remedy is ridging, which may be viewed as shrinkage of the local linear estimator towards the origin. In this paper we propose a general form of shrinkage, and suggest that, in practice, shrinkage be towards a proper curve estimator. For the latter we propose a local linear estimator based on an infinitely supported kernel. This approach is resistant against selection of too large a shrinkage parameter, which can impair performance when shrinkage is towards the origin. It also removes problems of numerical instability resulting from using a compactly supported kernel, and enjoys very good mean squared error properties.     

VARIATIONAL BAYESIAN ANALYSIS FOR HIDDEN MARKOV MODELS

The variational approach to Bayesian inference enables simultaneous estimation of model parameters and model complexity. An interesting feature of this approach is that it also leads to an automatic choice of model complexity. Empirical results from the analysis of hidden Markov models with Gaussian observation densities illustrate this. If the variational algorithm is initialized with a large number of hidden states, redundant states are eliminated as the method converges to a solution, thereby leading to a selection of the number of hidden states. In addition, through the use of a variational approximation, the deviance information criterion for Bayesian model selection can be extended to the hidden Markov model framework. Calculation of the deviance information criterion provides a further tool for model selection, which can be used in conjunction with the variational approach.     

On perfect simulation for some mixtures of distributions

This paper studies the implementation of the coupling from the past (CFTP) method of Propp and Wilson (1996) in the set-up of two and three component mixtures with known components and unknown weights. We show that monotonicity structures can be exhibited in both cases, but that CFTP can still be costly for three component mixtures. We conclude with a simulation experiment exhibiting an almost perfect sampling scheme where we only consider a subset of the exhaustive set of starting values.     

Perfect samplers for mixtures of distributions

Summary. We consider the construction of perfect samplers for posterior distributions associated with mixtures of exponential families and conjugate priors, starting with a perfect slice sampler in the spirit of Mira and co-workers. The methods rely on a marginalization akin to Rao–Blackwellization and illustrate the duality principle of Diebolt and Robert. A first approximation embeds the finite support distribution on the latent variables within a continuous support distribution that is easier to simulate by slice sampling, but we later demonstrate that the approximation can be very poor. We conclude by showing that an alternative perfect sampler based on a single backward chain can be constructed. This alternative can handle much larger sample sizes than the slice sampler first proposed.     

A RETROSPECTIVE INVESTIGATION OF GENDER INEQUALITY AND FEMALE HOMICIDE VICTIMIZATION

ABSTRACT

For the period 1981 to 1993 women represented 21 to 26 percent of homicide victims in the United States annually (Smith and Kuchta 1993 Smith , M. Dwayne and Ellen S. Kuchta . 1993 . “Trends in Violent Crime Against Women, 1973–89.” Social Science Quarterly 74 : 28 – 45 . [CSA]  [Google Scholar]). During this same time period sex-specific homicide rates have been among the forms of disaggregation researchers have used to test the utility of traditional correlates of homicide in predicting rates across various population subgroups and units of analysis. Based upon earlier research of the effects of gender inequality upon rates of lethal violence against women, and by applying a feminist theoretical perspective, it is hypothesized that the effect of general social structural characteristics of cities upon women's risk of homicide is mediated by levels of gender inequality. Specifically, this study examines the effects of gender, socioeconomic, legislative, political and extra-legal inequality upon female homicide victimization among 217 U.S. central cities for the period of 1989–1991. Using structural equation modeling results indicate that, among traditional social structural factors, economic deprivation, population size, divorce rate, and the sex ratio all have significant, positive effects on female homicide rates. However, in subsequent models testing the mediating effects of measures of gender inequality on the association between social structural variables and female homicide rates the divorce rate is the only social structural factor that continues to have a significant, positive effect upon homicide rates. Among the four measures of gender inequality, and in support of an ameliorative feminist argument, socioeconomic inequality has a significant, positive influence on rates of female homicide victimizaton. There is also a significant, negative effect of gender legislative inequality upon these rates. That is, the more laws or acts favorable to women, the lower their rates of homicide victimization. Implications of these findings are discussed.     

An algorithm for optimal designs on a design space

Two variations of a simple monotunic algorithm for computing optimal designs on a finite design space are presented. Various properties are listed. Comparisons witn other algorithms are made.     

Variational Bayes for estimating the parameters of a hidden Potts model

Hidden Markov random field models provide an appealing representation of images and other spatial problems. The drawback is that inference is not straightforward for these models as the normalisation constant for the likelihood is generally intractable except for very small observation sets. Variational methods are an emerging tool for Bayesian inference and they have already been successfully applied in other contexts. Focusing on the particular case of a hidden Potts model with Gaussian noise, we show how variational Bayesian methods can be applied to hidden Markov random field inference. To tackle the obstacle of the intractable normalising constant for the likelihood, we explore alternative estimation approaches for incorporation into the variational Bayes algorithm. We consider a pseudo-likelihood approach as well as the more recent reduced dependence approximation of the normalisation constant. To illustrate the effectiveness of these approaches we present empirical results from the analysis of simulated datasets. We also analyse a real dataset and compare results with those of previous analyses as well as those obtained from the recently developed auxiliary variable MCMC method and the recursive MCMC method. Our results show that the variational Bayesian analyses can be carried out much faster than the MCMC analyses and produce good estimates of model parameters. We also found that the reduced dependence approximation of the normalisation constant outperformed the pseudo-likelihood approximation in our analysis of real and synthetic datasets.     

Domestic violence and South Korean women: the cultural context and alternative experiences

The present research contributes to the growing body of cross-cultural research on domestic violence. This is accomplished by answering the question of how severity of intimate partner abuse varies for (1) women incarcerated for the homicides of their male partners (2) abused women who sought domestic violence shelter, short of killing their intimate assailants, and (3) a group of South Korean females outside of domestic violence shelters or prison. The article concludes with a discussion of potential policy implications of the findings as well as promising directions for future research.     

Hierarchical Gaussian process mixtures for regression

As a result of their good performance in practice and their desirable analytical properties, Gaussian process regression models are becoming increasingly of interest in statistics, engineering and other fields. However, two major problems arise when the model is applied to a large data-set with repeated measurements. One stems from the systematic heterogeneity among the different replications, and the other is the requirement to invert a covariance matrix which is involved in the implementation of the model. The dimension of this matrix equals the sample size of the training data-set. In this paper, a Gaussian process mixture model for regression is proposed for dealing with the above two problems, and a hybrid Markov chain Monte Carlo (MCMC) algorithm is used for its implementation. Application to a real data-set is reported.