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.     

The Siable-Law Model of Stock Returns

This study investigates the tail shapes of empirical distributions of returns on an extensive group of common stocks. The tails of the return distributions are found to be thinner than those of infinite variance stable distributions. Therefore, although homogeneity is evident in general, economic and statistical inferences drawn from stable-law parameters estimated from samples of stock returns may be misleading. This is in spite of the apparent overall similarity (in shape) between empirical and stable distributions.     

Confidence Intervals for the Weighted Sum of Two Independent Binomial Proportions

Confidence intervals for the difference of two binomial proportions are well known, however, confidence intervals for the weighted sum of two binomial proportions are less studied. We develop and compare seven methods for constructing confidence intervals for the weighted sum of two independent binomial proportions. The interval estimates are constructed by inverting the Wald test, the score test and the Likelihood ratio test. The weights can be negative, so our results generalize those for the difference between two independent proportions. We provide a numerical study that shows that these confidence intervals based on large‐sample approximations perform very well, even when a relatively small amount of data is available. The intervals based on the inversion of the score test showed the best performance. Finally, we show that as for the difference of two binomial proportions, adding four pseudo‐outcomes to the Wald interval for the weighted sum of two binomial proportions improves its coverage significantly, and we provide a justification for this correction.     

A model averaging approach for the ordered probit and nested logit models with applications

This paper considers model averaging for the ordered probit and nested logit models, which are widely used in empirical research. Within the frameworks of these models, we examine a range of model averaging methods, including the jackknife method, which is proved to have an optimal asymptotic property in this paper. We conduct a large-scale simulation study to examine the behaviour of these model averaging estimators in finite samples, and draw comparisons with model selection estimators. Our results show that while neither averaging nor selection is a consistently better strategy, model selection results in the poorest estimates far more frequently than averaging, and more often than not, averaging yields superior estimates. Among the averaging methods considered, the one based on a smoothed version of the Bayesian Information criterion frequently produces the most accurate estimates. In three real data applications, we demonstrate the usefulness of model averaging in mitigating problems associated with the ‘replication crisis’ that commonly arises with model selection.     

On the role of statistics in the palaeomagnetic proof of continental drift

Magnetized rocks provide a “tape recording” of the more recent history of the earth. It was this evidence that broke the opposition to continental drift. The nature of the data - samples of unit vectors in three dimensions - demanded new forms of statistical treatment. I hope that my personal account will enable the general statistician to see a rare example of statistics being developed to be a modest help in a major scientific revolution.     

The weighted compounding of probabilities from three independent hypothesis tests

Tail probabilities from three independent hypothesis tests can be combined to form a test statistic of the form P₁,P₂ ^θ2,P₃ ^θ3.The null distribution of the combined test statistic is presented and critical values for α=0.01 and 0.05 are provided.The power of this test is discussed for the special case ofthree independent F-tests.     

Maintaining family stability in the age of digital technologies: An analysis of d/Discourse informing domestic screen media practices in three US families

This article examines d/Discourses about screen media and parenting as reflected in parents' discursive construction of their decision-making processes connected with domestic media practices. The analysis focuses on three parent interviews from a dataset of 51 parent interviews conducted in three states in the USA. The article illustrates ways d/Discourses connected with health, education, ethnicity and disability that are specific to these three households intersect with d/Discourses concerning children and media. The article argues that general categories describing parenting practices overlook the microcultures of households, erasing the emotional labour embedded in processes of decision-making.     

Inventory management and cost-benefit analysis: A case study

This paper treats the problem of comparing various stocking and issuing policies for two radioactive isotopes, I-123 and I-131, which are used in nuclear medicine departments for the diagnosis of thyroid disease. The approach is based on using a combination of a cost-benefit analysis to compare the increased operating costs of stocking I-123 versus the increased disease risk of using I-131, and simulation to recreate the operation of a radio-pharmacy which stocks these drugs. The required inputs to the model include: (1) estimates of the disease risks associated with radiation absorbed doses from I-131; (2) estimates of the probability distributions of the appropriate latent periods; (3) empirical distributions of daily patient demand; (4) census data for estimating longevity and individual expected earnings and (5) cost data the stocking costs of the isotopes. The cost benefit results give issuing rules based on the age and sex of the patient, while the simulation is used to compare the effectiveness of the various ordering policies considered (the choice of a particular ordering policy may influence the risk of disease development).