Finite mixtures of multivariate skew t-distributions: some recent and new results

Finite mixtures of multivariate skew t (MST) distributions have proven to be useful in modelling heterogeneous data with asymmetric and heavy tail behaviour. Recently, they have been exploited as an effective tool for modelling flow cytometric data. A number of algorithms for the computation of the maximum likelihood (ML) estimates for the model parameters of mixtures of MST distributions have been put forward in recent years. These implementations use various characterizations of the MST distribution, which are similar but not identical. While exact implementation of the expectation-maximization (EM) algorithm can be achieved for ‘restricted’ characterizations of the component skew t-distributions, Monte Carlo (MC) methods have been used to fit the ‘unrestricted’ models. In this paper, we review several recent fitting algorithms for finite mixtures of multivariate skew t-distributions, at the same time clarifying some of the connections between the various existing proposals. In particular, recent results have shown that the EM algorithm can be implemented exactly for faster computation of ML estimates for mixtures with unrestricted MST components. The gain in computational time is effected by noting that the semi-infinite integrals on the E-step of the EM algorithm can be put in the form of moments of the truncated multivariate non-central t-distribution, similar to the restricted case, which subsequently can be expressed in terms of the non-truncated form of the central t-distribution function for which fast algorithms are available. We present comparisons to illustrate the relative performance of the restricted and unrestricted models, and demonstrate the usefulness of the recently proposed methodology for the unrestricted MST mixture, by some applications to three real datasets.     

Flexible mixture modelling using the multivariate skew-t-normal distribution

This paper presents a robust probabilistic mixture model based on the multivariate skew-t-normal distribution, a skew extension of the multivariate Student’s t distribution with more powerful abilities in modelling data whose distribution seriously deviates from normality. The proposed model includes mixtures of normal, t and skew-normal distributions as special cases and provides a flexible alternative to recently proposed skew t mixtures. We develop two analytically tractable EM-type algorithms for computing maximum likelihood estimates of model parameters in which the skewness parameters and degrees of freedom are asymptotically uncorrelated. Standard errors for the parameter estimates can be obtained via a general information-based method. We also present a procedure of merging mixture components to automatically identify the number of clusters by fitting piecewise linear regression to the rescaled entropy plot. The effectiveness and performance of the proposed methodology are illustrated by two real-life examples.     

A review of the theories of corporate social responsibility: Its evolutionary path and the road ahead

This study aims to trace the conceptual evolutionary path of theories on corporate social responsibility (CSR) and to reflect on the implications of the development. The retrospection has revealed that the trend has been a progressive rationalization of the concept with a particular focus on tighter coupling with organizations’ financial goals. Rationalization involves two broad shifts in the conceptualization of CSR. First, in terms of the level of analysis, researchers have moved from the discussion of the macro‐social effects of CSR to organizational‐level analysis of CSR's effect on profit. Next, in terms of theoretical orientation, researchers have moved from explicitly normative and ethics‐oriented arguments to implicitly normative and performance‐oriented managerial studies. Based on the retrospection, the limitations of the current state of CSR research that places excessive emphasis on the business case for CSR are outlined, and it is suggested that future research needs to refocus on basic research in order to develop conceptual tools and theoretical mechanisms that explain changing organizational behavior from a broader societal perspective.     

Assessing influence in time series

This paper studies influential observations on the spectrum of a stationary stochastic process. We introduce a leave-one-out procedure in spectral density estimation to identify influential points. A simulated envelope is proposed to assess the magnitude of influence when the data follow an autoregressive integrated moving average model. Practical illustrations are discussed in two examples.     

Semiparametric Estimators for Limited Dependent Variable (LDV) Models with Endogenous Regressors

This article reviews semiparametric estimators for limited dependent variable (LDV) models with endogenous regressors, where nonlinearity and nonseparability pose difficulties. We first introduce six main approaches in the linear equation system literature to handle endogenous regressors with linear projections: (i) ‘substitution’ replacing the endogenous regressors with their projected versions on the system exogenous regressors x, (ii) instrumental variable estimator (IVE) based on E{(error) × x} = 0, (iii) ‘model-projection’ turning the original model into a model in terms of only x-projected variables, (iv) ‘system reduced form (RF)’ finding RF parameters first and then the structural form (SF) parameters, (v) ‘artificial instrumental regressor’ using instruments as artificial regressors with zero coefficients, and (vi) ‘control function’ adding an extra term as a regressor to control for the endogeneity source. We then check if these approaches are applicable to LDV models using conditional mean/quantiles instead of linear projection. The six approaches provide a convenient forum on which semiparametric estimators in the literature can be categorized, although there are a few exceptions. The pros and cons of the approaches are discussed, and a small-scale simulation study is provided for some reviewed estimators.     

Time-of-Day Electricity Consumption Response to Temperature and the Ownership of Air Conditioning Appliances

This article is concerned with the endogeneity of air conditioning appliance ownership in the consumption of electricity by time of day (TOD). Both TOD consumption and air conditioning appliance ownership depend on temperature level and on household responsiveness to temperature variation, as well as on other measured household characteristics. The article presents an appropriate econometric methodology and applies it to obtain estimates and perform tests based on data from Southern California Edison's Residential TOD Pricing Experiment. The authors emphasize the implications for estimated consumption price elasticities.     

Stratification of randomization is not required for a pre‐specified subgroup analysis

Published literature and regulatory agency guidance documents provide conflicting recommendations as to whether a pre‐specified subgroup analysis also requires for its validity that the study employ randomization that is stratified on subgroup membership. This is an important issue, as subgroup analyses are often required to demonstrate efficacy in the development of drugs with a companion diagnostic. Here, it is shown, for typical randomization methods, that the fraction of patients in the subgroup given experimental treatment matches, on average, the target fraction in the entire study. Also, mean covariate values are balanced, on average, between treatment arms in the subgroup, and it is argued that the variance in covariate imbalance between treatment arms in the subgroup is at worst only slightly increased versus a subgroup‐stratified randomization method. Finally, in an analysis of variance setting, a least‐squares treatment effect estimator within the subgroup is shown to be unbiased whether or not the randomization is stratified on subgroup membership. Thus, a requirement that a study be stratified on subgroup membership would place an artificial roadblock to innovation and the goals of personalized healthcare. Copyright © 2012 John Wiley & Sons, Ltd.