Anant P. Godbole

t-Covering arrays: Upper bounds and poisson approximations

A k × n array with entries from the q-letter alphabet {0, 1, . . . , q − 1} is said to be t-covering if each k × t submatrix has (at least one set of) q distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n, t, q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let Kλ = Kλ(n, t, q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the q possible “words” of length t at least λ times. The Lovász lemma yields an upper bound on Kλ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k×n array, the Stein-Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e., missing at least one word.

POISSON APPROXIMATIONS FOR RUNS AND PATTERNS OF RARE EVENTS

Consider a sequence of Bernoulli trials with success probability p, and let N,,k denote the number of success runs of length k ?_i 2 among the first n trials. The Stein-Chen method is employed to obtain a total variation upper bound for the rate of convergence of N,,k to a Poisson random variable under the standard condition

Specific formulae for some success run distributions

Let N(k)n denote the number of success runs of length k ([greater-or-equal, slanted] 1) in n Bernoulli trials. A specific formula is derived for P(N(k)n = x) which is alternative to the one established by Philippou and Makri (1986) and Hirano (1986) and which is in a form suitable for the computation of asymptotic distributions (as in Godbole, 1990a, b); recall that N(k)n is said to have a binomial distribution of order k. In a similar fashion, different formulae are obtained for the geometric, negative binomial and Poisson distributions of order k (introduced by Philippou, Georghiou and Philippou, 1983.

A New Exact Runs Test for Randomness

Let Nn,k denote the number of recurrent success runs of length k≥2 in a sample of size n drawn with replacement from a dichotomous population. The exact distribution of Nn,k has recently been obtained in closed algorithmically simple form; we discuss the programming of these algorithms for values of n that are large, but not so large that asymptotic results can be invoked. Using the conditional distribution of Nn,k we derive a test for randomness and compare it with standard procedures based on runs, ranks, and variances. The simulation results showed that the new test is significantly more powerful in detecting certain types of clustering. Applications in neurology and reliability are provided.

The exact and asymptotic distribution of overlapping success runs

Let M = Mn,k and N = Nn,k denote, respectively, the number of overlapping and non-overlapping success runs of fixed length k in n Bernoulli (p) trials. We derive an alternative formula for P(M = x) which is substantially simpler to apply than the one in Ling (1988). Our method of proof brings out several connections between the distributions of M and N. Poisson approximations and a Berry-Esseen theorem are obtained for M, and applications are indicated.

Improved upper bounds for the reliability of d-dimensional consecutive-k-out-of-n : F systems

Consider a 2-dimensional consecutive-k-out-of-n : F system, as described by Salvia and Lasher [9], whose components have independent, perhaps identical, failure probabilities. In this paper, we use Jansons exponential inequalities [5]; to derive improved upper bounds on such a systems reliability, and compare our results numerically to previously determined upper bounds. In the case of equal component-failure probabilities, we determine analytically, given k and n, those component-failure probabilities for which our bound betters the upper bounds found by Fu and Koutras [4] and Koutras et al. [6]. A different kind of analytic comparison is made with the upper bound of Barbour et al. [3]. We further generalize our upper bound, given identical component-failure probabilities, to suit d-dimensional systems for d ≤ 3.

On hypergeometric and related distributions of order k

Let Nn.k.g.d be the hypergeometric random variable of order k≥1, equal to the number of success runs of length k contained in an ordered without replacement sample of size n drawn from a dichotomous urn with g good items and d defectives. We give an alternative formula for that is computationally simpler than the one in Panaretos and Xekalaki (1986). Distributions of the longest success run and of waiting times for r≥1 runs of length k are also derived. We call the latter the waiting time hypergeometric r.v. of order.

Degenerate and poisson convergence criteria for success runs

Let N(k)n be the number of success runs of length k > 1 in n Bernoulli trials, each with success probability pn. We show that N(k)n converges weakly to the distribution degenerate at zero as n --> [infinity], nf(pn) --> [lambda] (0 [infinity]). This answers, in the negative, a question posed by Philippou and Makri (1986) who suspected that a Poisson distribution of order k might be the target limit (if [is proportional to](pn) = pn). If, instead, npkn --> [lambda], we prove that N(k)n tends in law to a Poisson([lambda]) random variable. This improves a classical result of von Mises (1921) which required, in addition, that k --> [infinity]. Rates of convergence are provided for the above results.

On Universal Cycles for new Classes of Combinatorial Structures

A universal cycle (u-cycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of u-cycles for collections of subsets, restricted multisets, and lattice paths. For subsets, we show that a u-cycle exists for the

Note: Improved pebbling bounds

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