Herbert Solomon

A Gaussian Approximation to the Distribution of a Definite Quadratic Form

Abstract Let Qk = Σk j-1 cj(xj + aj)2 be a definite quadratic form in independent standardized Gaussian variables, xj, EQk = 01. The normalizing transformation (Qk/01)h is investigated, where h is determined by the first three moments of Qk. A new Gaussian approximation to the noncentral chi-square (x2) distribution is found for which the coefficient of skewness is smaller than a cube root transformation in the literature. Our transformation specializes to the cube root transformation of Wilson and Hilferty for the central x2 distribution. The approximation is simple to apply and compares well with other approximations in a number of cases studied numerically.

Distribution of a Sum of Weighted Chi-Square Variables

Abstract We consider distributions of quadratic forms of the type Qk = Σ k j = 1 cj (xj + aj )2, where the xj s are independent and identically distributed standard normal variables, and where cj and aj are nonnegative constants. Exact significance points of Qk , for selected values of cj and all aj = 0, have been published for k = 2, 3, 4, and 5. We give significance points for k = 6, 8, and 10. We propose and assess two new approximations to Qk : (1) fitting a Pearson curve with the same first four moments as Qk ; and (2) fitting Qk = Awr , where w has the χ2 p distribution, and where A, r, and p are determined by the first three moments of Qk.

Approximations to Density Functions Using Pearson Curves

Abstract This article is an expository paper to demonstrate the usefulness of Pearson curves in density estimation especially for those unaware of this early development in statistics. It is shown how to fit the curves and how very good approximate percentage points can be obtained for intractable distributions when the first four moments (or three moments and one endpoint) are known exactly (not estimated from sample data). The effectiveness of this method in density estimation is illustrated in three somewhat disparate contexts and reference is given to others. In general, the Pearson curves give an excellent approximation to the long tail of a distribution, the tail most often needed in practical work.

ON RANDOM SEQUENTIAL PACKING IN THE PLANE AND A CONJECTURE OF PALASTI.

Abstract : A conjecture by Palasti on mean values for random packing is explored. After thorough experimental examination and with several theoretical results as anchor points, the conjecture is demonstrated to be incorrect. The mean packing density in two dimension is a little higher than the square of the mean density in one dimension. The conjecture for higher dimensions remains an open question but some work is presented to provide clues. Other issues arise in random packing of points on a lattice and these are developed. (Author)

A second-order approximation for the variance of a renewal reward process

Let {C(t), t [greater-or-equal, slanted] 0} be a renewal reward process. We obtain the approximation Var C(t) = ct + d + o(1), and explicitly identify c and d.

A Study of Poisson's Models for Jury Verdicts in Criminal and Civil Trials

Abstract This article gives an exposition of models provided by Poisson to account for actual jury decisions in criminal and civil trials in France in the first half of the 19th century. The model parameters are jury size, juror error, and probability that the accused is guilty. Poissons models are cast in a modern setting, and this provides a framework in which new investigations can be initiated. A recent decision by the U.S. Supreme Court on jury size for felony proceedings in state courts should provoke more analyses that can build on the basis provided.

Modeling Jury Verdicts in the American Legal System

Abstract Jury size, majorities required for acquittal or conviction and correctness of juror decisions in criminal trials are studied. The article draws heavily on ideas presented in a previous article [1] which updated a model suggested by Poisson. Briefly, a model is developed and then examined, employing data from Kalven and Zeisel [2] on the American legal system. Comparisons of the American experience with French data from the early 19th century are made through the use of the model. Recent U. S. Supreme Court decisions in criminal trials regarding jury size and the relaxation of unanimity for decisions in criminal trials make these studies pertinent.

Optimal issuing policies under stochastic field lives

Abstract : Consider a stockpile of n items where the ith item has a rating (r sup i), i = 1,...,n. An item with rating (r sup i) if released to the field at time 0 will have a random field life distributed as x sub (r sub i); X sub (r sub i), i = 1,...,n is a collection of random variables which are increasing in r in the sense of monotone likelihood ratio. An item with rating (r sup i) if kept in stockpile until time t and then released to the field will have a field life distributed as X sub (r sub i)d(t) where d(t) is a non-random function. Items are to be issued from the stockpile to the field until the stockpile is depleted. The ith issued item is placed in the field immediately upon the death in the field of the (i-1)st issued item. The problem studied is to find the order of item issue which maximizes in some sense the total field life obtained from the stockpile. (Author)

Analyzing the Decision-Making Process of the American Jury

Abstract Extending earlier work by the authors [3, 4], this article examines in more detail the internal and external behavior of a jury. The internal analysis sheds light on the decision-making process from the first ballot position to the final outcome. Data in Kalven and Zeisel [6] have been employed to test the tenability of the models. The external analysis explores questions regarding the correctness of jury decisions and the effect of changes in jury size and in majority requirements for decisions.

Optimal Design of Sampling from Finite Populations: A Critical Review and Indication of New Research Areas

Abstract The design of sampling from finite populations is receiving increased attention in the statistical literature. This article presents a critical review of past studies and new developments and outlines areas meriting further study and research. Bayesian and non-Bayesian approaches are analyzed but are not offered in a normative manner. Both points of view and the extent to which they provide differences in survey sample design are discussed and examined. These differences are highlighted in stratification procedures, stratified sampling allocation, multivariate designs, and designs with varying inclusion probabilities.