Military Statistics† Herbert Solomon, Herbert SolomonSearch for more papers by this authorEdward J. Wegman, Edward J. WegmanSearch for more papers by this author Herbert Solomon, Herbert SolomonSearch for more papers by this authorEdward J. Wegman, Edward J. WegmanSearch for more papers by this author First published: 17 November 2014 https://doi.org/10.1002/9781118445112.stat00099 †This article was originally published online in 2006 in Encyclopedia of Statistical Sciences, © John Wiley & Sons, Inc. and republished in Wiley StatsRef: Statistics Reference Online, 2014. Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Wiley StatsRef: Statistics Reference OnlineBrowse other articles of this reference work:BROWSE BY TOPICBROWSE A-Z RelatedInformation

Multidimensional Wilson-Hilferty transformations support Gaussian approximations to certain joint distributions of quadratic forms in jointly Gaussian variates. Central and noncentral joint distributions are studied and applications are noted. Parameters of the approximating distributions are given up to terms of order o(nu-2) in the degrees of freedom nu. Numerical studies validate using these approximations over a range of parameters for an essential subclass of the distributions studied, especially in upper tails.

Tables are given for the distribution of a statistic for testing equality of variances where the intent is to discover if one variance is an outlier. These tables extend previously published tables. There is an obvious application to process control. It has become increasingly the practice to examine the variance as well as the mean in samples taken at different time points to check on the stability of the process. These tables should be helpful to the practitioner.

Exact expressions for the distribution function of a random variable of the form c 1 χ 2 m +c 2 χ 2 n are given where χ 2 m and χ 2 n χ 2 n are independent chi-square random variables with m and n degrees of freedom respectively. (The positive c i are distinct). In particular, the exact asymptotic distribution function for the average Kendall tau statistic is written as a function of tables of Solomon (1960) and some found in Abramowitz and Stegun's Handbook of Mathematical Functions.

The tables give probabilities of quardratic forms of the form where the positive a sum to one and X2 are independent chi-square random variables each with one degree of freedom and n is 2 or 3. The text provides formulas for missing values when n=2 relating them to the non-central chi-square tables of HYNAM, GOVINDARAJULU LEONE and SIEFERT. It also give simple formulas for the probabilities of quadratic forms with larger values of n when some of the coefficients a are alike. They enable one to use either these tables or those of JOHNSON and KOTZ (for n=4 or 5) to compute the probabilities. Specific examples of the use of the fomulas are provided

This paper describes the computer simulation of a coverage problem in geometric probability, that of placing random caps on the surface of a sphere. The simulation results were compared with exact values, where known, and the differences were negligible. This suggested the use of simulation results to assess several approximation formulas in the literature.

The random pazking problem has been of interest to inoestigsccrs in seveal disciplines , Physical chemists have investigaced such models in two and three dimensions, Because of aralytical difficulties, one-dimensional analogacs have been explored and theseare referred to as the parking problem, A number of results areexplored and attempts are made to tie them together, Applicationsare also highlighted.

At City College in New York City, I took one course in mathematical statistics and three courses in what today we term noncalculus statistics. The former was given by the department of mathematics in which I was enrolled to fulfill the requirement for a major subject. Professor Selby Robinson was my instructor, as he was for generations of students. The noncalculus statistics courses were not listed in any specific department and, in fact, were labeled Unattached 15.1, 15.2, and 15.3. They were offered by Professor John Firestone, who also provided statistical instruction to countless students over the years. Among my contemporaries at City College, (1936–40) were a number of individuals such as Kenneth J. Arrow, Herman Chernoff and Milton Sobel who made their mark in statistics and allied subjects in later years.

This note gives estimates of solutions to the Sylvester problem in three dimensions. These results suggest that the solutions in four and higher dimensions are very close to the solution for the n -dimensional sphere which is known.

AbstractSuggestions are put forth regarding a hedonic measure of well‐being based on individual freedom of choice and similarity of group behavior in the market. A type of backward segmentation using multivariate analysis is described. The suggested measure appears justifiable as an indexing device, a constant relative measure of well‐being, and/or a gauge of poverty‐wealth thresholds.

The analysis of jury size and jury verdicts in criminal m a t t e r s now has along , though interrupted , history . Work In this subject in the 18th and 19th centuries by Condorcet arid Laplace is discussed and the Poisson model of the 1830's is highighted . The latter is modified t o analyze the America1 jury experience of the 20th century. Recent U. S. Supreme Court decis on sin the 1970's on jury size and jury decisiol - making have created a resurgence of interest especially on a comparison of six member and twelve member juries . Some comparisons of size in terms of probabilities of errors invericts are presented.

Percentage points are given for Neyman's smooth goodness-of-fit statistic of order two. Recent work has indicated that this is a good omnibus test statistic for uniformity.

This chapter discusses the selection of representative points in normal populations. The representation of a continuous random variable by several discrete points occurs often in applied probability problems. Quantization is the term applied to this procedure. For the one-dimensional normal random variable situation, the univariate normal is quantized in an optimal manner. Maximizing the correlation is equivalent to minimizing mean square error. The nonoptimality of the symmetric quantizer is a feature of an odd number of points. The chapter discusses some special cases of two- and three-dimensional random variables.

AbstractA simple three‐moment approximation is introduced for the distribution of the sample variance. Comparisons are given with other approximations discussed by Tan and Wong (1977) and with an approximation developed very recently by Mudholkar and Trivedi (1981).

The problem of estimating the variance of a finite population is studied in a Bayesian framework. On the basis of the moderntheoretical approach to sampling from finite populations and the special structure of the likelihood functions Bayes estimators of the population variance are derived. The structure of equivariant estimators is analyzed and Bayes equivariant estimators in the strict and the generalized sense are derived. The Bayes risk efficiency of the classical estimators is studied

Many random variables arising in problems of geometric probability have intractable densities, and it is very difficult to find probabilities or percentage points based on these densities. A simple approximation, a generalization of the chi-square distribution, is suggested, to approximate such densities; the approximation uses the first three moments. These may be theoretically derived, or may be obtained from Monte Carlo sampling.The approximation is illustrated on random variables (the area, the perimeter, and the number of sides) associated with random polygons arising from two processes in the plane. Where it can be checked theoretically, the approximation gives good results. It is compared also with Pearson curve fits to the densities.

We have considered before the problem of estimating the parameters giving the mean time in each stage, for a two-stage Poisson process, when sampling was permitted only at equal intervals. It was impossible to get good results unless the intervals were small. We now propose an adaptive strategy in which the interval is successively halved until a suitable stage is reached; then all samples can be combined to give estimates. The strategy is examined by Monte Carlo methods, and it is shown to give a considerable improvement over the one-stage method. Figures are given to illustrate the results; they can be used also to improve estimates and give confidence intervals. We propose a technique to give an approximate confidence ellipse for the two parameters, which works well for the ranges considered.