One fifth (4 of 20) of the research articles published in the Journal of Educational Statistics in 1988 include simulation studies that justify or illustrate the authors' conclusions. A similar fraction (6 of 33) of the articles in the 1988 volume of Psychometrika include simulations; comparable proportions could be expected in other journals at the boundary of theoretical statistics and social/psychological applications. Due in part to the complexity of the problems tackled today and in part to the availability of cheap, powerful computing—by no means independent influences—simulation and Monte Carlo methods have become both necessary and practical tools for statisticians and applied workers in quantitative areas of education and psychology. Simulation has become popular—not only in the quantitative social sciences, but in all of the mathematical sciences from physics to operations research to number theory—because it is almost always easy to do. This ease of use makes the simulation experimenter vulnerable to two common pitfalls. Selection of the basic source of "random numbers" is often passive: Whatever is available in the computer's standard subroutine library is used. However, the fact that a pseudo-random number generator appears in a popular software package or operating system is hardly reason to trust it, as is shown by the infamous RANDU generator, once popular on IBM mainframes and PDP mini-computers, and by the generators burned into RAM on today's PCs. Simulation design and reporting also deserve special care. Some attempt must be made to assess the accuracy of the simulation estimates: One should accurately estimate and report SE (6) as well as 6. In addition, enough detail should be reported that the interested reader can replicate the study and check the results, just as with other experiments. Yet these considerations are also easy to overlook. Brian D. Ripley's Stochastic Simulation is a short, yet ambitious, survey of modern simulation techniques. Three themes run throughout the book. First, one shoud not take basic simulation subroutines for granted, especially on minior microcomputers where they tend to be poor implementations, implementations of poor algorithms, or both. Second, design of experiments, or variance reduction as it is known in this field, deserves greater consideration. Third, modern methods make it possible to simulate and analyze processes that are dependent over time, and using such processes opens the door to new simulation techniques, such as simulated annealing in optimization. Ripley intends this book to be a "comprehensive guide," and it is indeed most accurately described as a researcher's handbook with examples and

Stochastic Simulation

A comprehensive account of economic size distributions around the world and throughout the years In the course of the past 100 years, economists and applied statisticians have developed a remarkably diverse variety of income distribution models, yet no single resource convincingly accounts for all of these models, analyzing their strengths and weaknesses, similarities and differences. Statistical Size Distributions in Economics and Actuarial Sciences is the first collection to systematically investigate a wide variety of parametric models that deal with income, wealth, and related notions. Christian Kleiber and Samuel Kotz survey, compliment, compare, and unify all of the disparate models of income distribution, highlighting at times a lack of coordination between them that can result in unnecessary duplication. Considering models from eight languages and all continents, the authors discuss the social and economic implications of each as well as distributions of size of loss in actuarial applications. Specific models covered include: Pareto distributions Lognormal distributions Gamma-type size distributions Beta-type size distributions Miscellaneous size distributions Three appendices provide brief biographies of some of the leading players along with the basic properties of each of the distributions. Actuaries, economists, market researchers, social scientists, and physicists interested in econophysics will find Statistical Size Distributions in Economics and Actuarial Sciences to be a truly one-of-a-kind addition to the professional literature.

Statistical Size Distributions in Economics and Actuarial Sciences

In this paper, a new method is proposed for generating families of continuous distributions. A random variable $$X$$
 , “the transformer”, is used to transform another random variable $$T$$
 , “the transformed”. The resulting family, the $$T$$
 -
 $$X$$
 family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function of the random variable $$X$$
 . Many new distributions, which are members of the family, are presented. Several known continuous distributions are found to be special cases of the new distributions.

/pdf/a-new-method-for-generating-families-of-continuous-2e75jsuwp2.pdf

A new method for generating families of continuous distributions

The Hill equation was first introduced by A.V. Hill to describe the equilibrium relationship between oxygen tension and the saturation of haemoglobin. In pharmacology, the Hill equation has been extensively used to analyse quantitative drug-receptor relationships. Many pharmacokinetic-pharmacodynamic models have used the Hill equation to describe nonlinear drug dose-response relationships. Although the Hill equation is widely used, its many properties are not all well known. This article aims at reviewing the various properties of the Hill equation. The descriptive aspects of the Hill equation, in particular mathematical and graphical properties, are examined, and related to Hill's original work. The mechanistic aspect of the Hill equation, involving a strong connection with the Guldberg and Waage law of mass action, is also described. Finally, a probabilistic view of the Hill equation is examined. Here, we provide some new calculation results, such as Fisher information and Shannon entropy, and we introduce multivariate probabilistic Hill equations. The main features and potential applications of this probabilistic approach are also discussed. Thus, within the same formalism, the Hill equation has many different properties which can be of great interest for those interested in mathematical modelling in pharmacology and biosciences.

The Hill equation: a review of its capabilities in pharmacological modelling

The paper derives a function that describes the size distribution of incomes. The two functions most often used are the Pareto and the lognormal. The Pareto function fits the data fairly well towards the higher levels but the fit is poor towards the low income levels. The lognormal fits the lower income levels better but its fit towards the upper end is far from satisfactory. There have been other distributions suggested by Champernowne, Rutherford, and others, but even these do not result in any considerable improvement. The present paper derives a distribution that is a generalization of the Pareto distribution and the Weibull distribution used in analyses of equipment failures. The distribution fits actual data remarkably well compared with the Pareto and the lognormal.

/pdf/a-function-for-size-distribution-of-incomes-2qoewno20h.pdf

A Function for Size Distribution of Incomes

https://projecteuclid.org/download/pdf_1/euclid.aoms/1177731607

Cumulative Frequency Functions

/pdf/calculation-of-exact-sampling-distribution-of-ranges-from-a-2ndhzlj2o9.pdf

Irving W. Burr

Papers

Cumulative Frequency Functions

Calculation of Exact Sampling Distribution of Ranges from a Discrete Population