We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.

We are thankful to the discussants for their hard, interesting work. The main purpose of our paper was to give reasonably sharp rates of convergence for some simple examples of the Gibbs sampler. We chose examples from expository accounts where direct use of available techniques gave practically useless answers. Careful treatment of these simple examples grew into bivariate modeling and Lancaster families. Since bounding rates of convergence is our primary focus, let us begin there.

Model selection and assessment with incomplete data pose challenges in addition to the ones encountered with complete data. There are two main reasons for this. First, many models describe characteristics of the complete data, in spite of the fact that only an incomplete subset is observed. Direct comparison between model and data is then less than straightforward. Second, many commonly used models are more sensitive to assumptions than in the complete-data situation and some of their properties vanish when they are fitted to incomplete, unbalanced data. These and other issues are brought forward using two key examples, one of a continuous and one of a categorical nature. We argue that model assessment ought to consist of two parts: (i) assessment of a model’s fit to the observed data and (ii) assessment of the sensitivity of inferences to unverifiable assumptions, that is, to how a model described the unobserved data given the observed ones.

In randomized experiments, treatment and control groups should be roughly the same—balanced—in their distributions of pretreatment variables. But how nearly so? Can descriptive comparisons meaningfully be paired with significance tests? If so, should there be several such tests, one for each pretreatment variable, or should there be a single, omnibus test? Could such a test be engineered to give easily computed p-values that are reliable in samples of moderate size, or would simulation be needed for reliable calibration? What new concerns are introduced by random assignment of clusters? Which tests of balance would be optimal?

To address these questions, Fisher’s randomization inference is applied to the question of balance. Its application suggests the reversal of published conclusions about two studies, one clinical and the other a field experiment in political participation.

The logit model is often used to analyze experimental data. However, randomization does not justify the model, so the usual estimators can be inconsistent. A consistent estimator is proposed. Neyman’s non-parametric setup is used as a benchmark. In this setup, each subject has two potential responses, one if treated and the other if untreated; only one of the two responses can be observed. Beside the mathematics, there are simulation results, a brief review of the literature, and some recommendations for practice.

Current reporting of results based on Markov chain Monte Carlo computations could be improved. In particular, a measure of the accuracy of the resulting estimates is rarely reported. Thus we have little ability to objectively assess the quality of the reported estimates. We address this issue in that we discuss why Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they can be used to decide when to stop the simulation. We compare their use to a popular alternative in the context of two examples.

Karl Pearson played an enormous role in determining the content and organization of statistical research in his day, through his research, his teaching, his establishment of laboratories, and his initiation of a vast publishing program. His technical contributions had initially and continue today to have a profound impact upon the work of both applied and theoretical statisticians, partly through their inadequately acknowledged influence upon Ronald A. Fisher. Particular attention is drawn to two of Pearson’s major errors that nonetheless have left a positive and lasting impression upon the statistical world.

Jayaram Sethuraman was born in the town of Hubli in Bombay Province (now Karnataka State) on October 3, 1937. His early years were spent in Hubli and in 1950 his family moved to Madras (now renamed Chennai). He graduated from Madras University in 1957 with a B.Sc. (Hons) degree in statistics and he earned his M.A. degree in statistics from Madras University in 1958. He earned a Ph.D. in statistics from the Indian Statistical Institute in 1962. Before returning to ISI in 1965 as an Associate Professor, he was a Research Associate at the University of North Carolina 1962–1963, at Michigan State University in 1963–1964 and at Stanford University 1964–1965. After three years at the ISI, Sethuraman moved to Florida State University in 1968 as Full Professor. During his career at FSU, he made sojourns as Visiting Professor to the University of Michigan, 1974–1975, the ISI in fall 1977, as a Visiting Professor and Acting Head, ISI Bangalore Center, 1979–1980. He was a senior ASA/NSF/NIST Fellow 1994–1995 and a Fulbright Senior Researcher at ISI Bangalore 1995–1996.

Although Sethuraman officially retired on January 31, 2004 and was named Professor Emeritus at FSU, he continues to be extremely active. He participates in all activities in the Department of Statistics and holds a Courtesy Professor appointment in the Department of Religion. He held an appointment as Professor, University of Pittsburgh in the fall of 2004, and was a Fulbright Senior Lecturer at the Indian Statistical Institute of Technology, Chennai, 2005.

Sethuraman has been a superior researcher throughout his career, making important contributions in many areas including asymptotic distribution theory, large deviations theory, moderate deviations theory for which he was the pioneer, limit theory, nonparametric statistics, Dirichlet processes and Bayesian nonparametrics, stopping times for sequential estimation and testing, order statistics, stochastic majorization, Bahadur and Pitman efficiency, Markov chain Monte Carlo, reliability theory, survival analysis and image analysis. Throughout his career, he has enjoyed continuous external research support from the U.S. Army Office of Research and support from the Academy of Applied Science for mentoring high school students.

Jayaram Sethuraman has received many recognitions for his contributions to the discipline of statistics and to the advancement of science among future scholars. He was elected Fellow of the Institute of Mathematical Statistics (1968) and the American Statistical Association (1971), and became an elected member of the International Statistical Institute (1972). He received the U.S. Army S. S. Wilks Award (1994), was the R. A. Bradley Lecturer, University of Georgia (1995), received the Teaching Incentive Program Award, FSU (1995), and the Professorial Excellence Award, FSU (1996). He was chairman of the FSU Statistics Department 1987–1990. Sethuraman received an ASA Service Award (2001), the President’s Continuing Education Award, FSU (2002), and the Bhargavi and C. R. Rao Prize, Pennsylvania State University (2005). In 1993 he was named the Robert O. Lawton Distinguished Professor, FSU. This award is made to only one faculty member per year and is the University’s highest faculty honor.