Bernard Harris

Statistical Inference in the Classical Occupancy Problem Unbiased Estimation of the Number of Classes

A random sample of N observations is obtained from a multinomial distribution with an unknown number θ of equiprobable cells. The existence and asymptotic properties of the unbiased estimator based on the number of occupied cells is given. The related problem in which observations are made sequentially, stopping after L repeated cells have been observed, is also discussed. In the fixed sample size case, if θ is known not to exceed N, the unbiased estimator is a ratio of Stirling numbers of the second kind. For the sequential ease, the unbiased estimator is always a ratio of Stirling numbers of the second kind.

Hypothesis Testing and Confidence Intervals for Products and Quotients of Poisson Parameters with Applications to Reliability

Abstract X1 X2, ···, Xk1, Y1, Y2, ···, Yk2 are k1 + k2 mutually independent Poisson random variables with parameters λ1, λ2, ···, λk1, μ1, μ2, ···, μk2, respectively. Confidence intervals and tests of hypotheses for the parameter θ = λ1λ2 ··· λk1 / μ1μ2 ··· μk2 are obtained. Under suitable conditions these procedures may be used to obtain approximate confidence intervals and tests of hypotheses of the parameter ρ = ρ1ρ2 ··· ρk1/ρk1+1ρk1+2 ··· ρk1+k2, where the ρis, i = 1, 2, ···, k1 + k2 are binomial parameters. This problem is of importance in reliability analysis and some applications to reliability analysis are exhibited.

The use of the tetrachoric series for evaluating multivariate normal probabilities

The tetrachoric series is a technique for evaluating multivariate normal probabilities frequently cited in the statistical literature. In this paper we have examined the convergence properties of the tetrachoric series and have established the following. For orthant probabilities, the tetrachoric series converges if ;[varrho]ij; 1/(k - 1) or k is odd and [varrho]ij > 1/(k - 2), 1 = 2 and all [varrho]ij such that the correlation matrix is positive definite is false.

Theory and counterexamples for confidence limits on system reliability

The general theory of optimal confidence limits for system reliability based on results of Buehler (1957) and Sudakov and references cited there (1974) is developed. Counterexamples to some results of Sudakov (1974), Winterbottom (1974) and Harris and Soms (1980, 1981) are exhibited. Numerical examples are given for two component series systems which show that the results hold for confidence levels used in practice.

A note on the asymptotic normality of the distribution of the number of empty cells in occupancy problems

Abstract : Two elementary proofs of the asymptotic normality of the distribution of the number of empty cells in occupancy problems are provided. (Author)

The Asymptotic Distribution of the Order of Elements in Symmetric Semigroups.

Abstract P. Erdos and P. Turan [8] (Acta Math. Acad. Sci. Hungar., 18 (1967) , 309–320) have shown that, if K(n, x) is the number of elements P in Sn, the symmetric group on n letters, whose order O(P) satisfies log O(P) ⩽ 1 2 log 2 n + ( 1 3 ) x log 3 2 n then lim n→∞ K(n,x) n! = ( 2π ) −1 ∫ x −∞ e −t 2 2 dt. In this paper the analogous result for the symmetric semigroup is obtained. Let α ϵ Tn, the symmetric semigroup on n letters (the set of all mappings of {1, 2,…, n} into {1, 2,…, n}) and let O(α) be the order of α. If L(n, x) is the number of α ϵ Tn with log O(α)⩽ 1 8 log 2 n +(− 1 24 ) x log 3 2 n, then lim n→∞ L(n,x) n n = ( 2π ) −1 ∫ x ∞ e −t 2 2 dt.

A generalization of the Eulerian numbers with a probabilistic application

In this paper we study a generalization of the Eulerian numbers and a class of polynomials related to them. An interesting application to probability theory is given in Section 3. There we use these extended Eulerian numbers to construct an uncountably infinite family of lattice random variables whose first n moments coincide with the first n moments of the sum of n+1 uniform random variables. A number of combinatorial identities are also deduced.

Properties of the Generalized Incomplete Modified Bessel Distributions with Applications to Reliability Theory

Abstract In B. Harris [8], a family of distributions useful in statistical inference on the reliability of systems of independent parallel components was introduced. In the present article properties of this family of distributions are explored. The moments are obtained and asymptotic expressions for the mean and variance are given. Asymptotic normality is exhibited and relations to a number of special functions are indicated. Various asymptotic results and limit theorems are obtained and some applications to reliability theory are also given.

The Distribution of Linear Combinations of the Sample Occupancy Numbers.

Assume that a random sample of n observations has been taken from the multinomial distribution with N equiprobable cells. Let si, i = 0, 1, 2, …, n, be the number of cells occurring i times in the sample. Let W=∑i=0nwisi be a linear combination of the sis. In this paper we obtain a sufficient condition under which the asymptotic distribution of W is normal as n and N tend to infinity such that n/N → α>0.

The Distribution of the Number of Empty Cells in a Generalized Random Allocation Scheme.

n balls are randomly distributed into N cells, so that no cell may contain more than one ball. This process is repeated m times. In addition, balls may disappear; such disappearances are independent and identically Bernoulli distributed. Conditions are given under which the number of empty cells has an asymptotically ( N →∞) standard normal distribution.