Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the virtues of the ellipse and her higher-dimensional cousins for both these purposes in a variety of contexts, including linear models, multivariate linear models and mixed-effect models. We emphasize the strong relationships among statistical methods, matrix-algebraic solutions and geometry that can often be easily understood in terms of ellipses.

The problem of small area estimation (SAE) is how to produce reliable estimates of characteristics of interest such as means, counts, quantiles, etc., for areas or domains for which only small samples or no samples are available, and how to assess their precision. The purpose of this paper is to review and discuss some of the new important developments in small area estimation methods. Rao [Small Area Estimation (2003)] wrote a very comprehensive book, which covers all the main developments in this topic until that time. A few review papers have been written after 2003, but they are limited in scope. Hence, the focus of this review is on new developments in the last 7–8 years, but to make the review more self-contained, I also mention shortly some of the older developments. The review covers both design-based and model-dependent methods, with the latter methods further classified into frequentist and Bayesian methods. The style of the paper is similar to the style of my previous review on SAE published in 2002, explaining the new problems investigated and describing the proposed solutions, but without dwelling on theoretical details, which can be found in the original articles. I hope that this paper will be useful both to researchers who like to learn more on the research carried out in SAE and to practitioners who might be interested in the application of the new methods.

Based on the principles of importance sampling and resampling, sequential Monte Carlo (SMC) encompasses a large set of powerful techniques dealing with complex stochastic dynamic systems. Many of these systems possess strong memory, with which future information can help sharpen the inference about the current state. By providing theoretical justification of several existing algorithms and introducing several new ones, we study systematically how to construct efficient SMC algorithms to take advantage of the “future” information without creating a substantially high computational burden. The main idea is to allow for lookahead in the Monte Carlo process so that future information can be utilized in weighting and generating Monte Carlo samples, or resampling from samples of the current state.

We prove a theorem justifying the regularity conditions which are needed for Path Sampling in Factor Models. We then show that the remaining ingredient, namely, MCMC for calculating the integrand at each point in the path, may be seriously flawed, leading to wrong estimates of Bayes factors. We provide a new method of Path Sampling (with Small Change) that works much better than standard Path Sampling in the sense of estimating the Bayes factor better and choosing the correct model more often. When the more complex factor model is true, PS-SC is substantially more accurate. New MCMC diagnostics is provided for these problems in support of our conclusions and recommendations. Some of our ideas for diagnostics and improvement in computation through small changes should apply to other methods of computation of the Bayes factor for model selection.

We present a framework for selecting and developing measures of dependence when the goal is the quantification of a relationship between two variables, not simply the establishment of its existence. Much of the literature on dependence measures is focused, at least implicitly, on detection or revolves around the inclusion/exclusion of particular axioms and discussing which measures satisfy said axioms. In contrast, we start with only a few nonrestrictive guidelines focused on existence, range and interpretability, which provide a very open and flexible framework. For quantification, the most crucial is the notion of interpretability, whose foundation can be found in the work of Goodman and Kruskal [Measures of Association for Cross Classifications (1979) Springer], and whose importance can be seen in the popularity of tools such as the $R^{2}$ in linear regression. While Goodman and Kruskal focused on probabilistic interpretations for their measures, we demonstrate how more general measures of information can be used to achieve the same goal. To that end, we present a strategy for building dependence measures that is designed to allow practitioners to tailor measures to their needs. We demonstrate how many well-known measures fit in with our framework and conclude the paper by presenting two real data examples. Our first example explores U.S. income and education where we demonstrate how this methodology can help guide the selection and development of a dependence measure. Our second example examines measures of dependence for functional data, and illustrates them using data on geomagnetic storms.

Although fractional Brownian motion was not invented by Benoît Mandelbrot, it was he who recognized the importance of this random process and gave it the name by which it is known today. This is a personal account of the history behind fractional Brownian motion and some subsequent developments.