The goal of this paper is to investigate properties of statistical procedures based on numbers of different patterns by using generating functions for the probabilities of a prescribed number of occurrences of given patterns in a random text. The asymptotic formulae are derived for the expected value of the number of words occurring a given number of times and for the covariance matrix. The form of the optimal linear test based on these statistics is established. These problems appear in testing for the randomness of a string of binary bits, DNA sequencing, source coding, synchronization, quality control protocols, etc. Indeed, the probabilities of repeated (overlapping) patterns are important in information theory (the second-order properties of relative frequencies)and molecular biology problems (finding patterns with unexpectedly low or high frequencies).

Testing in order to produce software of high reliability is an area of major concern in software engineering. In an effort to find efficient methods of testing, the comparison of partition and random sampling testing methods has received considerable attention in the literature. A standard criterion for comparisons between random and partition testing, based on their expected efficacy in program debugging, is the probability of detecting at least one failure causing input in the program's domain. However, the goal in software testing is usually to find as many faults as possible in a reasonable period of time, and therefore stochastic comparisons of the number of faults obtained in partition and random testing may provide more valuable information on which testing procedures to use. We establish various conditions which guarantee that the number of faults discovered in partition testing is stochastically greater than the number discovered in random testing (using a fixed total sample size) for many of the well-established stochastic orders (including the usual stochastic order, the hazard rate order, the likelihood ratio order, and the variability order). The results established also allow us to obtain both upper and lower bounds with these stochastic orders for the sum of n independent Bernoulli random trials (with varying probability of success) in terms of the binomial distribution with parameters n and p.

We deal with compound geometric sums of independent positive random variables and study the moment problem for the distributions of such sums (the Stieltjes moment problem). We find conditions under which the distributions are uniquely determined by their moments. We also treat related topics, including the Hamburger moment problem involving random variables on the whole real line. Some conjectures are outlined.

In this paper, we clarify dependence properties of elliptical distributions by deriving general but explicit formulae for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendall's tau is invariant in the class of elliptical distributions with continuous marginals and a fixed dispersion matrix, we show that this is not true for Spearman's rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts.

In this paper we find conditions under which the epoch times of two nonhomogeneous Poisson processes are ordered in the multivariate dispersive order. Some consequences and examples of this result are given. These results extend a recent result of Brown and Shanthikumar (1998).

We extend classical renewal theorems to the weighted case. A hierarchical chain of successive sharpenings of asymptotic statements on the weighted renewal functions is obtained by imposing stronger conditions on the weighting coefficients.

Simple approximation techniques are developed exploiting relationships between generalized convex orders and appropriate probability metrics. In particular, the distance between s-convex ordered random variables is investigated. Results connecting positive or negative dependence concepts and convex ordering are also presented. These results lead to approximations and bounds for the distributions of sums of positively or negatively dependent random variables. Applications and extensions of the main results pertaining to compound Poisson, normal and exponential approximation are provided as well.

Ebrahimi (2001) proposes an interesting model for assessing a system's reliability, expressing the failure time of the system in terms of a deterioration process and covariates. He also provides illustrative examples and gives some properties of the model. In this note, we give conditions for negative ageing of the system's lifetime, and we correct one of his statements. Moreover, for the lifetimes of two systems of the same kind, some stochastic comparisons are presented.

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

This paper first recalls some stochastic orderings useful for studying the ℒ-class and the Laplace order in general. We use these orders to show that the higher moments of an ℒ-class distribution need not exist. Using simple sufficient conditions for the Laplace ordering, we give examples of distributions in the ℒ- and ℒα-classes. Moreover, we present explicit sharp bounds on the survival function of a distribution belonging to the ℒ-class of life distributions. The results reveal that the ℒ-class should not be seen as a more comprehensive class of ageing distributions but rather as a large class of life distributions on its own.

In this note, we give new proofs of the closure property of ageing classes NBUC and NBU(2) under convolution to make up the gaps in the proofs of Cao and Wang (1991) and of Li and Kochar (2001).

Ease of computation of Fréchet-optimal lower bounds, given numerical values of the binomial moments S ij , i, j = 1, 2, is demonstrated. A sufficient condition is given for an explicit bivariate bound of Dawson-Sankoff structure to be Fréchet optimal. An example demonstrates that in the bivariate case even the multiplicative structure of the S ij does not guarantee a Dawson-Sankoff structure for Fréchet-optimal bounds. A final section is used to illuminate the nature of Fréchet optimality by using generalized explicit bounds. This note is a sequel to both Chen and Seneta (1995) and Chen (1998).

Let be the scan statistic of window size r for a sequence of n bistate trials . The scan statistic S n (r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of S n (r) are rather complex, and they can only be numerically implemented when is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of S n (r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.

Products of independent identically distributed random stochastic 2 × 2 matrices are known to converge in distribution under a trivial condition. Rates for this convergence are estimated in terms of the minimal L p -metrics and the Kolmogoroff metric and applications to convergence rates of related interval splitting procedures are discussed.

In this paper we utilize a particular transformation of i.i.d. exponential random variables to derive two distributional identities. Throughout the analysis we discover some peculiar properties of exponentials. We also discuss possible generalizations and applications of the results.

During DNA replication, small fragments of DNA are formed. These have been observed experimentally and the mechanism of their formation modelled mathematically. Using the stochastic model of Cowan and Chiu (1992), (1994), we find the probability distribution of the number of fragments. A new discrete distribution arises. The work has interest as an application of the recent theory on quasirenewal equations in Piau (2000).

The key theme is converse forms of criteria for deciding determinateness in the classical moment problem. A method of proof due to Koosis is streamlined and generalized giving a convexity condition under which moments satisfying implies that c a positive constant. A contrapositive version is proved under a rapid variation condition on f (x), generalizing a result of Lin. These results are used to obtain converses of the Stieltjes versions of the Carleman and Krein criteria. Hamburger versions are obtained which relax the symmetry assumption of Koosis and Lin, respectively. A sufficient condition for Stieltjes determinateness of a discrete law is given in terms of its mass function. These criteria are illustrated through several examples.

Univariate probability inequalities have received extensive attention. It has been shown that under certain conditions, product-type bounds are valid and sharper than summation-type bounds. Although results concerning multivariate inequalities have appeared in the literature, product-type bounds in a multivariate setting have not yet been studied. This note explores an approach using graph theory and linear programming techniques to construct product-type lower bounds for the probability of the intersection among unions of k sets of events.

Intrinsic volumes are key functionals in convex geometry and have also appeared in several stochastic settings. Here we relate them to questions of regularity in Gaussian processes with regard to Itô–Nisio oscillation and metrization of GB/GC indexing sets. Various bounds and estimates are presented, and questions for further investigation are suggested. From alternate points of view, much of the discussion can be interpreted in terms of (i) random sets and (ii) properties of (deterministic) infinite-dimensional convex bodies.

Let X n , n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let X r,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of X r,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.