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Ranked preference data arise when a set of judges rank, in order of their preference, a set of objects. Such data arise in preferential voting systems and market research surveys. Covariate data associated with the judges are also often recorded. Such covariate data should be used in conjunction with preference data when drawing inferences about judges.

To cluster a population of judges, the population is modeled as a collection of homogeneous groups. The Plackett-Luce model for ranked data is employed to model a judge's ranked preferences within a group. A mixture of Plackett- Luce models is employed to model the population of judges, where each component in the mixture represents a group of judges.

Mixture of experts models provide a framework in which covariates are included in mixture models. Covariates are included through the mixing proportions and the component density parameters. A mixture of experts model for ranked preference data is developed by combining a mixture of experts model and a mixture of Plackett-Luce models. Particular attention is given to the manner in which covariates enter the model. The mixing proportions and group specific parameters are potentially dependent on covariates. Model selection procedures are employed to choose optimal models.

Model parameters are estimated via the ‘EMM algorithm’, a hybrid of the expectation–maximization and the minorization–maximization algorithms. Examples are provided through a menu survey and through Irish election data. Results indicate mixture modeling using covariates is insightful when examining a population of judges who express preferences.

Standard estimation of ARMA models in which the AR and MA roots nearly cancel, so that individual coefficients are only weakly identified, often produces inferential ranges for individual coefficients that give a spurious appearance of accuracy. We remedy this problem with a model that uses a simple mixture prior. The posterior mixing probability is derived using Bayesian methods, but we show that the method works well in both Bayesian and frequentist setups. In particular, we show that our mixture procedure weights standard results heavily when given data from a well-identified ARMA model (which does not exhibit near root cancellation) and weights heavily an uninformative inferential region when given data from a weakly-identified ARMA model (with near root cancellation). When our procedure is applied to a well-identified process the investigator gets the “usual results,” so there is no important statistical cost to using our procedure. On the other hand, when our procedure is applied to a weakly identified process, the investigator learns that the data tell us little about the parameters – and is thus protected against making spurious inferences. We recommend that mixture models be computed routinely when inference about ARMA coefficients is of interest.

We provide a Bayesian inference for a mixture of two Pareto distributions which is then used to approximate the upper tail of a wage distribution. The model is applied to the data from the CPS Outgoing Rotation Group to analyze the recent structure of top wages in the United States from 1992 through 2009. We find an enormous earnings inequality between the very highest wage earners (the “superstars”), and the other high wage earners. These findings are largely in accordance with the alternative explanations combining the model of superstars and the model of tournaments in hierarchical organization structure. The approach can be used to analyze the recent pay gaps among top executives in large firms so as to exhibit the “superstar” effect.

The problem of model mixing in time series, for which the interest lies in the estimation of stochastic volatility, is addressed using the approach known as Mixture-of-Experts (ME). Specifically, this work proposes a ME model where the experts are defined through ARCH, GARCH and EGARCH structures. Estimates of the predictive distribution of volatilities are obtained using a full Bayesian approach. The methodology is illustrated with an analysis of a section of US dollar/German mark exchange rates and a study of the Mexican stock market index using the Dow Jones Industrial index as a covariate.

The growth incidence curve of Ravallion and Chen (2003) is based on the quantile function. Its distribution-free estimator behaves erratically with usual sample sizes leading to problems in the tails. The authors propose a series of parametric models in a Bayesian framework. A first solution consists in modeling the underlying income distribution using simple densities for which the quantile function has a closed analytical form. This solution is extended by considering a mixture model for the underlying income distribution. However, in this case, the quantile function is semi-explicit and has to be evaluated numerically. The last solution consists in adjusting directly a functional form for the Lorenz curve and deriving its first-order derivative to find the corresponding quantile function. The authors compare these models by Monte Carlo simulations and using UK data from the Family Expenditure Survey. The authors devote a particular attention to the analysis of subgroups.

Studies on wage discrimination assume that independent observers are able to distinguish a priori which workers are suffering from discrimination. However, this may not be a good assumption when anti-discrimination laws mean that severe penalties can be imposed on discriminatory employers or when unobserved heterogeneity is significant. We develop a wage discrimination model in which workers are not classified a priori. It can be thought of as a generalization of the standard empirical framework, whereas the Oaxaca–Blinder model can be thought of as an extreme case. We propose a finite mixture model to explicitly model unobserved heterogeneity in individual characteristics and estimate the probabilities of being a discriminated or a non-discriminated worker. We illustrate this proposal by estimating wage discrimination in Germany and the UK.

This chapter proposes a new mixture model which allows for heterogeneity in sensitivities and decision rules across decision makers and attributes.

A new mixture model is put forward in which the different latent classes make use of different decision rules, where the use of generalised random regret minimisation kernel allows for within class heterogeneity in the decision rules applied across attributes.

Our theoretical developments are supported by the findings of an empirical application using data from a typical stated choice survey.

Existing work has looked at heterogeneity in decision rules and sensitivities across respondents. Other work has focused on the possibility that different decision rules apply to different attributes. This chapter puts forward a model that combines these two directions of research and does so in a way that lets the optimal specification be driven by the data rather than being imposed by the analyst.

Healthcare-associated infections (HAIs) are a major cause of concern because of the high levels of associated morbidity, mortality, and cost. In addition, children and intensive care unit (ICU) patients are more vulnerable to these infections due to low levels of immunity. Various medical interventions and statistical process control techniques have been suggested to counter the spread of these infections and aid early detection of an infection outbreak. Methods such as hand hygiene help in the prevention of HAIs and are well-documented in the literature. This chapter demonstrates the utilization of a systems methodology to model and validate factors that contribute to the risk of HAIs in a pediatric ICU. It proposes an approach that has three unique aspects: it studies the problem of HAIs as a whole by focusing on several HAIs instead of a single type, it projects the effects of interventions onto the general patient population using the system-level model, and it studies both medical and behavioral interventions and compares their effectiveness. This methodology uses a systems modeling framework that includes simulation, risk analysis, and statistical techniques for studying interventions to reduce the transmission likelihood of HAIs.

Alongside structural equation modeling (SEM), the complementary technique of partial least squares (PLS) path modeling helps researchers understand relations among sets of observed variables. Like SEM, PLS began with an assumption of homogeneity – one population and one model – but has developed techniques for modeling data from heterogeneous populations, consistent with a marketing emphasis on segmentation. Heterogeneity can be expressed through interactions and nonlinear terms. Additionally, researchers can use multiple group analysis and latent class methods. This chapter reviews these techniques for modeling heterogeneous data in PLS, and illustrates key developments in finite mixture modeling in PLS using the SmartPLS 2.0 package.