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Quasi-Monte Carlo (QMC) methods have begun to displace ordinary Monte Carlo (MC) methods in many practical problems. It is natural and obvious to combine QMC methods with traditional variance reduction techniques used in MC sampling, such as control variates. There can, however, be some surprises. The optimal control variate coefficient for QMC methods is not in general the same as for MC. Using the MC formula for the control variate coefficient can worsen the performance of QMC methods. A good control variate in QMC is not necessarily one that correlates with the target integrand. Instead, certain high frequency parts or derivatives of the control variate should correlate with the corresponding quantities of the target. We present strategies for applying control variate coefficients with QMC and illustrate the method on a 16-dimensional integral from computational finance. We also include a survey of QMC aimed at a statistical readership.
Ronald A. Fisher’s 1921 article on mathematical statistics (submitted and read in 1921; published in 1922) was arguably the most influential article on that subject in the twentieth century, yet up to that time Fisher was primarily occupied with other pursuits. A number of previously published documents are examined in a new light to argue that the origin of that work owes a considerable (and unacknowledged) debt to a challenge issued in 1916 by Karl Pearson.
In the past ten years there has been a dramatic increase of interest in the Bayesian analysis of finite mixture models. This is primarily because of the emergence of Markov chain Monte Carlo (MCMC) methods. While MCMC provides a convenient way to draw inference from complicated statistical models, there are many, perhaps underappreciated, problems associated with the MCMC analysis of mixtures. The problems are mainly caused by the nonidentifiability of the components under symmetric priors, which leads to so-called label switching in the MCMC output. This means that ergodic averages of component specific quantities will be identical and thus useless for inference. We review the solutions to the label switching problem, such as artificial identifiability constraints, relabelling algorithms and label invariant loss functions. We also review various MCMC sampling schemes that have been suggested for mixture models and discuss posterior sensitivity to prior specification.
Pareto, Zipf and numerous subsequent investigators of inverse power distributions have often represented their findings as though their data conformed to a power law form for all ranges of the variable of interest. I refer to this ideal case as a strong inverse power law (SIPL). However, many of the examples used by Pareto and Zipf, as well as others who have followed them, have been truncated data sets, and if one looks more carefully in the lower range of values that was originally excluded, the power law behavior usually breaks down at some point. This breakdown seems to fall into two broad cases, called here (1) weak and (2) false inverse power laws (WIPL and FIPL, resp.). Case 1 refers to the situation where the sample data fit a distribution that has an approximate inverse power form only in some upper range of values. Case 2 refers to the situation where a highly truncated sample from certain exponential-type (and in particular, “lognormal-like”) distributions can convincingly mimic a power law. The main objectives of this paper are (a) to show how the discovery of Pareto–Zipf-type laws is closely associated with truncated data sets; (b) to elaborate on the categories of strong, weak and false inverse power laws; and (c) to analyze FIPLs in some detail. I conclude that many, but not all, Pareto–Zipf examples are likely to be FIPL finite mixture distributions and that there are few genuine instances of SIPLs.
Shelley Zacks was born in Tel Aviv on October 15, 1932. He earned his B.A. degree in statistics, mathematics and sociology from Hebrew University in 1955, an M.Sc. degree in operations research and statistics from the Technion in 1960, and a Ph.D. degree in operations research from Columbia University in 1962. He is perhaps best known for his groundbreaking articles on change-point problems, common mean problems, Bayes sequential strategies and reliability analysis. His lifelong enthusiasm in handling difficult problems arising in science and engineering has been a primary inspiration behind his most important theoretical publications. His studies on survival probabilities in crossing mine fields as well as his contributions in stochastic visibility in random fields are regarded as fundamental work in naval research and other defense related areas. Professor Zacks’ authoritative book, The Theory of Statistical Inference (1971), and its 1975 Russian translation have served graduate programs and researchers all over the globe very well for over 30 years. He has written other books and monographs, including Parametric Statistical Inference:Basic Theory and Modern Approaches (1981b), Introduction to Reliability Analysis: Probability Models and Statistical Methods (1992), Prediction Theory for Finite Populations (1992), co-authored with H. Bolfarine, Stochastic Visibility in Random Fields (1994b) and Modern Industrial Statistics:Design and Control of Quality and Reliability (1998), co-authored with R. Kenet. He is the author or co-author of more than 150 research publications. During the period 1957 through 1980, his career path took him to the Technion (Israel Institute of Technology), New York University, Stanford University, Kansas State University, University of New Mexico, Tel Aviv University, Case Western Reserve University (CWRU) and Virginia Polytechnic Institute and State University (VPI). During 1974–1979, he was a Professor and Chairman of the Department of Mathematics and Statistics at CWRU. In 1979–1980, he spent a year in the Department of Statistics at VPI. In 1980 he moved to State University of New York–Binghamton (now called Binghamton University) as Professor and Chairman of the Department of Mathematical Sciences, and he has continued in the department as Professor and Director of the Center for Statistics, Quality Control and Design. For nearly 20 years, Professor Zacks worked as a consultant for the Program in Logistics at George Washington University. Professor Zacks has held a steady stream of editorial positions for such journals as Journal of the American Statistical Association, The Annals of Statistics, Journal of Statistical Planning and Inference, Naval Research Logistics Quarterly, Communications in Statistics and Sequential Analysis. He served as the Executive Editor for Journal of Statistical Planning and Inference during 1998–2000. He has earned many honors and awards, including Fellow of the Institute of Mathematical Statistics (1974), Fellow of the American Statistical Association (1974), Fellow of the American Association for the Advancement of Science (1982) and elected membership in the International Statistical Institute (1975). He regularly travels to scientific conferences as an invited participant, works harder than many half his age, and continues to inspire through his writings and uniquely affectionate presence.