We introduce a new class of lifetime distributions exhibiting a notion of positive ageing, called the ℳ-class, which is strongly related to the well-known ℒ-class. It is shown that distributions in the ℳ-class cannot have an undesirable property recently observed in an example of an ℒ-class distribution by Klar (2002). Moreover, it is shown how these and related classes of life distributions can be characterized by expected remaining lifetimes after a family of random times, thus extending the notion of NBUE. We give examples of ℳ-class distributions by using simple sufficient conditions, and we derive reliability bounds for distributions in this class.

In this paper, we consider the total variation distance between the distributions of two random sums S M and S N with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum S N if M and N have the same first moments.

In this paper, we investigate sooner and later waiting time problems for patterns S 0 and S 1 in multistate Markov dependent trials. The probability functions and the probability generating functions of the sooner and later waiting time random variables are studied. Further, the probability generating functions of the distributions of distances between successive occurrences of S 0 and between successive occurrences of S 0 and S 1 and of the waiting time until the rth occurrence of S 0 are also given.

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).

The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space - namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w 1), …, N(S,w d )), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O ((log n) / √n) in the i.i.d. case and O ((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

This paper proposes matrix variate generalization of Kummer-Beta family of distributions which has been studied recently by Ng and Kotz. This distribution is an extension of Beta distribution. Its characteristic function has been derived and it is shown that the distribution is orthogonally invariant. Some results on distribution of random quadratic forms have also been derived.

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.

In this paper, we investigate k-out-of-n systems with independent and identically distributed components. Some characterizations of the IFR(2), DMRL, NBU(2) and NBUC classes of life distributions are obtained in terms of the monotonicity of the residual life given that the (n-k)th failure has occurred at time t ≥ 0. These results complement those reported by Belzunce, Franco and Ruiz (1999). Similar conclusions based on the residual life of a parallel system conditioned by the (n-k)th failure time are presented as well.

Connections between classification and lumpability in the stochastic Hopfield model (SHM) are explored and developed. A simplification of the SHM's complexity based upon its inherent lumpability is derived. Contributions resulting from this reduction in complexity include: (i) computationally feasible classification time computations; (ii) a development of techniques for enumerating the stationary distribution of the SHM's energy function; and (iii) a characterization of the set of possible absorbing states of the Markov chain associated with the zero temperature SHM.

In this paper we provide new results about stochastic comparisons of the excess lifetime at different times of a renewal process when the interarrival times belong to several ageing classes. We also provide a preservation result for the new better than used in the Laplace transform order ageing class for series systems.

The paper describes a new numerical method for the calculation of noncrossing probabilities for arbitrary boundaries by a Poisson process. We find the method to be simple in implementation, quick and efficient - it works reliably for Poisson processes of very high intensity n, up to several thousand. Hence, it can be used to detect unusual features in the finite-sample behaviour of empirical process and trace it down to very high sample sizes. It also can be used as a good approximation for noncrossing probabilities for Brownian motion and Brownian bridge, in particular when the boundaries are not regular. As a numerical example we demonstrate the divergence of normalized Kolmogorov-Smirnov statistics from their prescribed limiting distributions (Eicker (1979), Jaeshke (1979)) for quite large n in contrast to very regular behaviour of statistics of Mason (1983). For the Brownian motion case we considered square-root, Daniels' (1969) and Grooneboom's (1989) boundaries.

We study the last passage time and its asymptotic distribution for minimum contrast estimators defined through the minimization of a convex criterion function based on U-functionals. This includes cases of non-smooth estimators for vector valued parameters. We also derive a Bahadur-type representation and the law of iterated logarithms for such estimators.

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.

Consider a sequence X 1,…,X n of independent random variables with the same continuous distribution and the event X i-r+1 < ⋯ < X i of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.

We examine how the binomial distribution B(n,p) arises as the distribution S n = ∑i=1 n X i of an arbitrary sequence of Bernoulli variables. It is shown that B(n,p) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases B(2,p) and B(3,p) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which S n follows B(n,p), given that S n-1 is not necessarily a binomial variable, are investigated. Several natural characterizations of B(n,p), including one which relates the binomial distributions and the Poisson process, are also given. These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility.

Some new results about the NBU(2) class of life distributions are obtained. Firstly, it is proved that the decreasing with time of the increasing concave ordering of the excess lifetime in a renewal process leads to the NBU(2) property of the interarrival times. Secondly, the NBU(2) class of life distributions is proved to be closed under the formation of series systems. Finally, it is also shown that the NBU(2) class is closed under convolution operation.

A probabilistic proof of a Stein factor bound is given, extending the Poisson case in Xia (1999).

In this paper, a statistic that has been introduced to test for space-time correlation is considered in a time series context. The null hypothesis is white noise; the alternative is any kind of continuous functional dependence. For an autoregressive process close to the null hypothesis, a bound on the distance between the distribution of the statistic and a Poisson distribution is proved, using the Stein-Chen method. The main difficulty in the proof is that the dependence in the time series is not locally restricted. The result implies asymptotically certain discrimination for a reasonable choice of the thresholds.