This book provides expository derivations for moments of a family of pseudo distributions, which is an extended family of distributions including the pseudo normal (PN) distributions recently proposed by the author. The PN includes the skew normal (SN) derived by A. Azzalini and the closed skew normal (CSN) obtained by A. Domínguez-Molina, G. González-Farías, and A. K. Gupta as special cases. It is known that the CSN includes the SN and other various distributions as special cases, which shows that the PN has a wider variety of distributions. The SN and CSN have symmetric and skewed asymmetric distributions. However, symmetric distributions are restricted to normal ones. On the other hand, symmetric distributions in the PN can be non-normal as well as normal. In this book, for the non-normal symmetric distributions, the term “kurtic normal (KN)” is used, where the coined word “kurtic” indicates “mesokurtic, leptokurtic, or platykurtic” used in statistics. The variety of the PN was made possible using stripe (tigerish) and sectional truncation in univariate and multivariate distributions, respectively. The proofs of the moments and associated results are not omitted and are often given in more than one method with their didactic explanations.

This book provides expository derivations for moments of a family of pseudo distributions, which is an extended family of distributions including the pseudo normal (PN) distributions recently proposed by the author. The PN includes the skew normal (SN) derived by A. Azzalini and the closed skew normal (CSN) obtained by A. Domínguez-Molina, G. González-Farías, and A. K. Gupta as special cases. It is known that the CSN includes the SN and other various distributions as special cases, which shows that the PN has a wider variety of distributions. The SN and CSN have symmetric and skewed asymmetric distributions. However, symmetric distributions are restricted to normal ones. On the other hand, symmetric distributions in the PN can be non-normal as well as normal. In this book, for the non-normal symmetric distributions, the term “kurtic normal (KN)” is used, where the coined word “kurtic” indicates “mesokurtic, leptokurtic, or platykurtic” used in statistics. The variety of the PN was made possible using stripe (tigerish) and sectional truncation in univariate and multivariate distributions, respectively. The proofs of the moments and associated results are not omitted and are often given in more than one method with their didactic explanations.

The text and accompanying CD-ROM develop step by step a modern approach to econometric problems. They are aimed at talented upper-level undergraduates, graduate students, and professionals wishing to acquaint themselves with the pinciples and procedures for information processing and recovery from samples of economic data. The text fully provides an operational understanding of a rich set of estimation and inference tools, including tradional likelihood based and non-traditional non-likelihood based procedures, that can be used in conjuction with the computer to address economic problems.

"These papers were presented and developed as expository talks at a summer-long workshop on Stein's method at Stanford's Department of Statistics in 1998."--P. iii.

Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world’s leading experts on Benford’s law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration. Beginning with the general theory, the contributors explain the prevalence of the bias, highlighting explanations for when systems should and should not follow Benford’s law and how quickly such behavior sets in. They go on to discuss important applications in disciplines ranging from accounting and economics to psychology and the natural sciences. The contributors describe how Benford’s law has been successfully used to expose fraud in elections, medical tests, tax filings, and financial reports. Additionally, numerous problems, background materials, and technical details are available online to help instructors create courses around the book. Emphasizing common challenges and techniques across the disciplines, this accessible book shows how Benford’s law can serve as a productive meeting ground for researchers and practitioners in diverse fields.

The book investigates the misapplication of conventional statistical techniques to fat tailed distributions and looks for remedies, when possible. Switching from thin tailed to fat tailed distributions requires more than "changing the color of the dress." Traditional asymptotics deal mainly with either n=1 or n=∞, and the real world is in between, under the "laws of the medium numbers"-which vary widely across specific distributions. Both the law of large numbers and the generalized central limit mechanisms operate in highly idiosyncratic ways outside the standard Gaussian or Levy-Stable basins of convergence. A few examples: - The sample mean is rarely in line with the population mean, with effect on "naïve empiricism," but can be sometimes be estimated via parametric methods. - The "empirical distribution" is rarely empirical. - Parameter uncertainty has compounding effects on statistical metrics. - Dimension reduction (principal components) fails. - Inequality estimators (Gini or quantile contributions) are not additive and produce wrong results. - Many "biases" found in psychology become entirely rational under more sophisticated probability distributions. - Most of the failures of financial economics, econometrics, and behavioral economics can be attributed to using the wrong distributions. This book, the first volume of the Technical Incerto, weaves a narrative around published journal articles.