Robert T. Smythe

Central limit theorems for urn models

We consider central limit theory for urn models in which balls are not necessarily replaced after being drawn, giving rise to negative diagonal entries in the generating matrix. Under conditions on the eigenvalues and eigenvectors, we give results both for the contents of the urn and the number of times balls of each type are drawn.

On the structure of random plane‐oriented recursive trees and their branches

This paper is an investigation of the structural properties of random plane-oriented recursive trees and their branches. We begin by an enumeration of these trees and some general properties related to the outdegrees of nodes. Using generalized Polya urn models we study the exact and limiting distributions of the size and the number of leaves in the branches of the tree. The exact distribution for the leaves in the branches is given by formulas involving second-order Eulerian numbers. A martingale central limit theorem for a linear combination of the number of leaves and the number of internal nodes is derived. The distribution of that linear combination is a mixture of normals with a beta distribution as its mixing density. The martingale central limit theorem allows easy determination of the limit laws governing the leaves in the branches. Furthermore, the asymptotic joint distribution of the number of nodes of outdegree 0, 1 and 2 is shown to be trivariate normal.

Analysis of quickselect : an algorithm for order statistics

We study QUICKSELECT a one-sided version of QUICKSORT suited for finding the order statistics of a sample. We identify procedures by which the moments of the number of comparisons can be found exactly under both assumptions that the order statistic in question is randomly chosen or fixed. The procedure is illustrated by finding the exact mean and variance for a randomly selected order statistic as well as the first few in the fixed case. The existence of an absolutely continuous infinitely divisible limit law with asymmetric left and right tails is demonstrated in the case of a randomly chosen order statistic. Some of these distributional properties carry over to the case of a very small fixed order statistic.

Asymptotic Joint Normality of Outdegrees of Nodes in Random Recursive Trees

We study the joint probability distribution of the number of nodes of outdegree 0, 1, and 2 in a random recursive tree. We complete the known partial list of exact means and variances for outdegrees up to two by obtaining exact combinatorial expressions for the remaining means, variances, and covariances. The joint probability distribution of the number of nodes of outdegree 0, 1, and 2 is shown to be asymptotically trivariate normal and the asymptotic covariance structure is explicitly determined. It is also shown how to extend the results (at least in principle) to obtain a limiting multivariate normal distribution for nodes of outdegree 0, 1, …, k.

Analysis of Boyer-Moore-Horspool string-matching heuristic

We investigate the probabilistic behavior of a string-matching heuristic used for searching for the occurrences of a pattern in a random text. Our investigation covers the two cases when the pattern itself is fixed or random. Under suitable normalization we show that the total search time is asymptotically normally distributed in the case of fixed pattern, whereas in the case of random pattern the distribution of the search time becomes a mixture of degenerate distributions. An instrumental recurrence equation is obtained by shifting the pattern within the text. To handle the sum of dependent random variables appearing in the recurrence, analytic methods based on the behavior of the shift generating function near its dominant singularity in the complex plane are devised to yield moment calculation and the asymptotic distributions. Adaptation of the standard central limit theorem under mixing conditions complements our analytic toolkit.

Probabilistic Analysis of MULTIPLE QUICK SELECT

Abstract. We investigate the distribution of the number of comparisons made by MULTIPLE QUICK SELECT (a variant of QUICK SORT for finding order statistics). By convergence in the Wasserstein metric space, we show that a limit distribution exists for a suitably normalized version of the number of comparisons. We characterize the limiting distribution by an inductive convolution and find its variance. We show that the limiting distribution is smooth and prove that it has a continuous density with unbounded support.

Statistical Tests with Historical Controls

Carcinogen bioassay involves exposure of laboratory animals to one or more levels of the test substance, with tumor occurrence rates in the treated groups evaluated relative to those in unexposed controls (Bickis and Krewski, 1985a). The untreated animals are maintained on test at the same time as the treated animals and housed under the same conditions in the same laboratory. Because the concurrent controls are thus similar to the exposed animals in all respects except for treatment with the test chemical, it is generally agreed that they constitute the most appropriate reference group against which to compare the exposed groups.

Robust Tests for Trend in Binomial Proportions

Recent modifications to the Cochran-Armitage statistic used to test for trend in binomial proportions in carcinogenicity bioassays for which a series of historical control data is available employ a beta distribution for the between study variation in the binomial response rate in the control group. In this paper, the use of robust distributions with heavier tails than the beta is proposed as a means of accommodating the uncertainty as to the actual historical distribution of the binomial response rate. The robust distributions are selected from within a class of mixed distributions using a r-minimax criterion to select the most appropriate value of the mixing proportion. These tests are shown to be more robust than the existing tests with respect to inclusion or exclusion of individual historical control data points.

H.R. 3137 and the search for national information policy

Abstract The efforts of other nations to develop comprehensive approaches to the development and application of information technology have given rise to concern in many quarters about the adequacy of the U.S. governments response to information issues. A bill, H.R. 3137, was introduced in the House of Representatives in April 1981 by Congressman George E. Brown, Jr., in an attempt to stimulate movement toward a coordinated development of U.S. information policies. Hearings on this legislation elicited a wide variety of opinion concerning how best to proceed toward this development. This paper surveys the testimony presented at the hearings and some recent actions taken by Congress and by the Reagan Administration on information issues.

On the joint behavior of types of coupons in generalized coupon collection

The ‘coupon collection problem’ refers to a class of occupancy problems in which j identical items are distributed, independently and at random, to n cells, with no restrictions on multiple occupancy. Identifying the cells as coupons, a coupon is ‘collected’ if the cell is occupied by one or more of the distributed items; thus, some coupons may never be collected, whereas others may be collected once or twice or more. We call the number of coupons collected exactly r times coupons of type r. The coupon collection model we consider is general, in that a random number of purchases occurs at each stage of collecting a large number of coupons; the sample sizes at each stage are independent and identically distributed according to a sampling distribution. The joint behavior of the various types is an intricate problem. In fact, there is a variety of joint central limit theorems (and other limit laws) that arise according to the interrelation between the mean, variance, and range of the sampling distribution, and of course the phase (how far we are in the collection processes). According to an appropriate combination of the mean of the sampling distribution and the number of available coupons, the phase is sublinear, linear, or superlinear. In the sublinear phase, the normalization that produces a Gaussian limit law for uncollected coupons can be used to obtain a multivariate central limit law for at most two other types — depending on the rates of growth of the mean and variance of the sampling distribution, we may have a joint central limit theorem between types 0 and 1, or between types 0, 1, and 2. In the linear phase we have a multivariate central limit theorem among the types 0, 1,…, k for any fixed k.