A Bayesian method for classification and discrimination

We discuss Bayesian analyses of traditional normal-mixture models for classification and discrimination. The development involves application of an iterative resampling approach to Monte Carlo inference, commonly called Gibbs sampling, and demonstrates routine application. We stress the benefits of exact analyses over traditional classification and discrimination techniques, including the ease with which such analyses may be performed in a quite general setting, with possibly several normal-mixture components having different covariance matrices, the computation of exact posterior classification probabilities for observed data and for future cases to be classified, and posterior distributions for these probabilities that allow for assessment of second-level uncertainties in classification.     

Bayesian density regression for discrete outcomes

We develop Bayesian models for density regression with emphasis on discrete outcomes. The problem of density regression is approached by considering methods for multivariate density estimation of mixed scale variables, and obtaining conditional densities from the multivariate ones. The approach to multivariate mixed scale outcome density estimation that we describe represents discrete variables, either responses or covariates, as discretised versions of continuous latent variables. We present and compare several models for obtaining these thresholds in the challenging context of count data analysis where the response may be over‐ and/or under‐dispersed in some of the regions of the covariate space. We utilise a nonparametric mixture of multivariate Gaussians to model the directly observed and the latent continuous variables. The paper presents a Markov chain Monte Carlo algorithm for posterior sampling, sufficient conditions for weak consistency, and illustrations on density, mean and quantile regression utilising simulated and real datasets.     

Mixture inverse Gaussian for unobserved heterogeneity in the autoregressive conditional duration model

In this paper, we assume that the duration of a process has two different intrinsic components or phases which are independent. The first is the time it takes for a trade to be initiated in the market (for example, the time during which agents obtain knowledge about the market in which they are operating and accumulate information, which is coherent with Brownian motion) and the second is the subsequent time required for the trade to develop into a complete duration. Of course, if the first time is zero then the trade is initiated immediately and no initial knowledge is required. If we assume a specific compound Bernoulli distribution for the first time and an inverse Gaussian distribution for the second, the resulting convolution model has a mixture of an inverse Gaussian distribution with its reciprocal, which allows us to specify and test the unobserved heterogeneity in the autoregressive conditional duration (ACD) model.

Our proposals make it possible not only to capture various density shapes of the durations but also easily to accommodate the behaviour of the tail of the distribution and the non monotonic hazard function. The proposed model is easy to fit and characterizes the behaviour of the conditional durations reasonably well in terms of statistical criteria based on point and density forecasts.     

Updating discriminant functions estimated from inverse gaussian populations

The problem of updating discriminant functions estimated from inverse Gaussian populations is investigated in situations when the additional observations are mixed (unclassified) or classified. In each case two types of discriminant functions, linear and quadratic, are considered. Using simulation experiments the performance of the updating procedures is evaluated by means of relative efficiencies.     

An Empirical Approach to Sufficient Similarity: Combining Exposure Data and Mixtures Toxicology Data

When assessing risks posed by environmental chemical mixtures, whole mixture approaches are preferred to component approaches. When toxicological data on whole mixtures as they occur in the environment are not available, Environmental Protection Agency guidance states that toxicity data from a mixture considered “sufficiently similar” to the environmental mixture can serve as a surrogate. We propose a novel method to examine whether mixtures are sufficiently similar, when exposure data and mixture toxicity study data from at least one representative mixture are available. We define sufficient similarity using equivalence testing methodology comparing the distance between benchmark dose estimates for mixtures in both data‐rich and data‐poor cases. We construct a “similar mixtures risk indicator”(SMRI) (analogous to the hazard index) on sufficiently similar mixtures linking exposure data with mixtures toxicology data. The methods are illustrated using pyrethroid mixtures occurrence data collected in child care centers (CCC) and dose‐response data examining acute neurobehavioral effects of pyrethroid mixtures in rats. Our method shows that the mixtures from 90% of the CCCs were sufficiently similar to the dose‐response study mixture. Using exposure estimates for a hypothetical child, the 95th percentile of the (weighted) SMRI for these sufficiently similar mixtures was 0.20 (i.e., where SMRI <1, less concern; >1, more concern).     

Estimation of gaussian mixtures with rotationally invariant covariance matrices

Homoscedastic and heteroscedastic Gaussian mixtures differ in the constraints placed on the covariance matrices of the mixture components. A new mixture, called herein a strophoscedastic mixture, is defined by a new constraint, This constraint requires the matrices to be identical under orthogonal trans¬formations, where different transformations are allowed for different matrices. It is shown that the M-step of the EM method for estimating the parameters of strophoscedastic mixtures from sample data is explicitly solvable using singular value decompositions. Consequently, the EM-based maximum likelihood estimation algorithm is as easily implemented for strophoscedastic mixtures as it is for homoscedastic and heteroscedastic mixtures. An example of a “noisy” Archimedian spiral is presented.     

Estimation of a discriminant function from a mixture of two inverse Weibull distributions

The classification of a random variable based on a mixture can be meaningfully discussed only if the class of all finite mixtures is identifiable. In this paper, we find the maximum-likelihood estimates of the parameters of the mixture of two inverse Weibull distributions by using classified and unclassified observations. Next, we estimate the nonlinear discriminant function of the underlying model. Also, we calculate the total probabilities of misclassification as well as the percentage bias. In addition, we investigate the performance of all results through a series of simulation experiments by means of relative efficiencies. Finally, we analyse some simulated and real data sets through the findings of the paper.     

Information matrix for a mixture of two inverse gaussian distributions

In this paper, Fisher information matrix about the five parameters ρ, μ_:1, μ², λ¹and λ²of a mixture of two Inverse Gaussian density functions is obtained. The Leguerre-Gauss quadrature formula is used to evaluate the essential integral on which the twenty five elements of the information matrix are based. Results involving the computation of the information about p are compared with those involving both the power series expansion and Simpson's method of integration. Laguerre-Gauss quadra-ture was found to lead to good approximations as compared with other methods. It was therefore chosen for the computations of the elements of the information matrix.     

Degeneracy in the Maximum Likelihood Estimation of Univariate Gaussian Mixtures for Grouped Data and Behaviour of the EM Algorithm

Abstract.  In the context of the univariate Gaussian mixture with grouped data, it is shown that the global maximum of the likelihood may correspond to a situation where a Dirac lies in any non-empty interval. Existence of a domain of attraction near such a maximizer is discussed and we establish that the expectation-maximization (EM) iterates move extremely slowly inside this domain. These theoretical results are illustrated both by some Monte-Carlo experiments and by a real data set. To help practitioners identify and discard these potentially dangerous degenerate maximizers, a specific stopping rule for EM is proposed.     

Tests based on equivariant estimators

Let ?⁽¹⁾ and ?⁽²⁾ be location-equivariant estimators of an unknown location parameter μ. It is shown that the test for H₀: μ ≤ μ₀ versus H_A : μ > μ₀ that rejects H₀ if ?⁽¹⁾ is large is uniformly more powerful than the one that rejects H₀ if ?⁽²⁾ is large if and only if ?⁽²⁾ is “more dispersed” than ?⁽¹⁾. A similar result is obtained for tests on scale using the star-shaped ordering. Examples are given.