Joseph Naus

The Distribution of the Size of the Maximum Cluster of Points on a Line

Abstract N points are independently drawn from the uniform distribution on (0, 1). Denote by E(n|N; p), the event: There exists a subinterval of (0, 1) of length p that contains at least n out of the N points. We find the probability, P(n|N; p), of E(n|N; p) for n>N/2 in terms of simple tabulated quantities.

Approximations for Distributions of Scan Statistics

Abstract Certain statistical applications deal with the extremal distributions of the number of points in a moving interval or window of fixed length. This article gives an approximation that is highly accurate for several of these distributions. Applications include the maximum cluster of points on a line or circle, multiple coverage by subintervals or subarcs of fixed size, the length of the longest success run in Bernoulli trials, and the generalized birthday problem.

Probabilities for a Generalized Birthday Problem

Abstract Let 1s and M- 0s be randomly arranged in a row. Saperstein [11] defines the random variable k* m to be the maximum number of 1s within any m consecutive positions in the arrangement, and finds Pr (k* m . This article derives the distribution of k*m, for all for M/m = L, L an integer. Simplified forms are given for M/m = L, k > .

Power Comparison of Two Tests of Non-Random Clustering

This paper presents two definitions of maximum cluster, that have sometimes been used to test for non-random clustering. We compare the power of the testa based on these statistics, and show that when the clustering interval is sufficiently small, one of the tests is more powerful for a wide class of alternative hypotheses. We show that this test is the generalized likelihood ratio test for an alternative hypothesis related to a particular type of non-random clustering.

Some Probabilities, Expectations and Variances for the Size of Largest Clusters and Smallest Intervals

Abstract Given N points independently drawn from the uniform distribution on (0, 1), let [ptilde] n be the size of the smallest interval that contains n out of the N points; let n p be the largest number of points to be found in any subinterval of (0, 1) of length p. This paper uses a result of Karlin, McGregor, Barton, and Mallows to determine the distribution of n p , for p = 1/k, k an integer. The paper gives simple determinations for the expectations and variances of [ptilde] n , for all fixed n > (N + 1)/2, and of n1/2. The distribution and expectation of n p are estimated and tabulated for the cases p = 0.1(0.1)0.9, N = 2(1)10.

The Distribution of the Logarithm of the Sum of Two Log-Normal Variates

Abstract Let Z 1, Z 2 be two independent, identically distributed random variables whose logarithms are normally distributed. We derive the generating function, expectation, and variance of the logarithm of the sum of Z 1 and Z 2. The expressions for the expectation and variance involve the sums of rapidly converging series. Converging upper and lower bounds to the expectation are given to indicate the number of terms in the series that need to be evaluated to yield a specified number of significant places.

Probabilities for the Size of Largest Clusters and Smallest Intervals

Abstract Given N points distributed at random on [0,1), let np be the size of the largest number of points clustered within an interval of length p. Previous work finds Pr (np≥n), for n>N/2, and for n≤N/2, p=1/L, L an integer. The formula for the case p=1/L is in terms of the sum of L×L determinants and is not computationally feasible for large L. The present paper derives such a computational formula.

Multiple clusters on the line

Given N events occurring over time, define an n:t cluster as n consecutive events all contained within an interval of length t. In this paper we derive the expectation, variance and approximate distribution of the number of n:t clusters. The results have applications in epidemiological studies of rare diseases.

A Probabilistic Model for Identifying Errors in Data Editing

Abstract Certain data screening systems incorporate large numbers of logical checks on data entering the system. When violated, these logical checks indicate that various combinations of variates are in error. This article provides a model for assigning a probability measure to identify variates in error when there is a simultaneous violation of a set of logical checks. For certain symmetry conditions, the measure is a reasonable approximation to the posterior probability that given a violation of a set of conditions, a variate is in error.

New recursive methods for scan statistic probabilities

Abstract Glaz and Naus (1991) and Naus (1982) developed tight bounds and sharp approximations for the distribution of the maximum of moving sums of independent and identically distributed integer-valued random variables. These distributions are of importance in a wide variety of applications. To apply the bounds and approximations certain quantities need to be evaluated. We derive new recursive methods to compute efficiently these necessary quantities. An important area of application of our results is testing the significance of regions of high net charge in DNA and Protein sequences. We apply our results to reanalyze a set of data on charges in the Epstein-Barr Virus.