In this paper we study the moment generating function order and the new better than used in the moment generating function order (NBUMG) life distributions. A closure property of this order under an independent random sum is deduced, and stochastic comparisons among the block replacement policy, the age replacement policy, the complete repair policy, and the minimal repair policy of an NBUMG component are investigated.

Let {X i }i=1 n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable S n,m to be the maximum number of 1s within any m consecutive trials in {X i }i=1 n . The random variable S n,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {X i }i=1 n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of S n,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

Let {X n , n ≥ 1} be an independent, identically distributed random sequence with each X n having the general error distribution. In this paper we derive the exact uniform convergence rate of the distribution of the maximum to its extreme value limit.

In this paper we obtain several mixture representations of the reliability function of the inactivity time of a coherent system under the condition that the system has failed at time t (> 0) in terms of the reliability functions of inactivity times of order statistics. Some ordering properties of the inactivity times of coherent systems with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors between systems.

In Reinert and Röllin (2009) a new approach - called the ‘embedding method’ - was introduced, which allows us to make use of exchangeable pairs for normal and multivariate normal approximations with Stein's method in cases where the corresponding couplings do not satisfy a certain linearity condition. The key idea is to embed the problem into a higher-dimensional space in such a way that the linearity condition is then satisfied. Here we apply the embedding to U-statistics as well as to subgraph counts in random graphs.

Asymptotic independence of the components of random vectors is a concept used in many applications. The standard criteria for checking asymptotic independence are given in terms of distribution functions (DFs). DFs are rarely available in an explicit form, especially in the multivariate case. Often we are given the form of the density or, via the shape of the data clouds, we can obtain a good geometric image of the asymptotic shape of the level sets of the density. In this paper we establish a simple sufficient condition for asymptotic independence for light-tailed densities in terms of this asymptotic shape. This condition extends Sibuya's classic result on asymptotic independence for Gaussian densities.

Consider a sequence of exchangeable or independent binary trials ordered on a line or on a circle. The statistics denoting the number of times an F-S string of length (at least)k 1 + k 2, that is, (at least)k 1 failures followed by (at least) k 2 successes in n such trials, are studied. The associated waiting time for the rth occurrence of an F-S string of length (at least) k 1 + k 2 in linearly ordered trials is also examined. Exact formulae, lower/upper bounds and approximations are derived for their distributions. Mean values and variances of the number of occurrences of F-S strings are given in exact formulae too. Particular exchangeable and independent sequences of binary random variables, used in applied research, combined with numerical examples clarify further the theoretical results.

We consider a nearly unstable, or near unit root, AR(1) process with regularly varying innovations. Two different approximations for the stationary distribution of such processes exist: a Gaussian approximation arising from the nearly unstable nature of the process and a heavy-tail approximation related to the tail asymptotics of the innovations. We combine these two approximations to obtain a new uniform approximation that is valid on the entire real line. As a corollary, we obtain a precise description of the regions where each of the Gaussian and heavy-tail approximations should be used.

We apply Stein's method for probabilistic approximation by a compound geometric distribution, with applications to Markov chain hitting times and sequence patterns. Bounds on our Stein operator are found using a complex analytical approach based on generating functions and Cauchy's formula.

A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.

A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequential k-out-of-n system, where the distribution of the remaining components' lifetimes is allowed to change after every failure of a component. In the present paper, general component lifetimes which can be influenced by failures are considered. The ordered failure times of these components can be used to extend the concept of sequential order statistics. In particular, a definition of sequential order statistics based on exchangeable components is proposed. By utilizing the system signature (cf. Samaniego (2007)), the distribution of the lifetime of a coherent system with failure-dependent exchangeable component lifetimes is shown to be given by a mixture of the distributions of sequential order statistics. Furthermore, some results on the joint distribution of sequential order statistics based on exchangeable components are given.

In this paper we are concerned with the equilibrium distribution ∏n of the nth element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a (2+α)th moment condition on the jump distributions, we establish a bound of order O(n -(α+1)/2√logn) on the difference between the point probabilities of ∏n and those of a translated Poisson distribution with the same variance. Except for the factor √logn, the result is as good as could be obtained in the simpler setting of sums of independent, integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is a C ∞-function.

In this paper we refine a Poisson limit theorem of Gnedenko and Kolmogorov (1954): we determine the error order of a Poisson approximation for sums of asymptotically negligible integer-valued random variables that converge in distribution to the Poisson law. As an application of our results, we investigate the case of the coupon collector's problem when the distribution of the collector's waiting time is asymptotically Poisson.

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.

Global weak continuity of M-functionals in a neighbourhood of the parametric distribution is established. This has implications for robustness of M-estimators vis a vis definitions put forward by Hampel. For instance the Tukey bisquare location estimator is robust on neighbourhoods of the parametric model, but the median is not.

Let F(x, θ) be a family of distribution functions indexed by θ ∈ Ω. If G(θ) is a distribution function on Ω H(x) = ƒohm; F(x, θ) dG(θ) is a mixture with respect to G. If there is a unique G yielding H, the mixtures is said to be identifiable.This paper summarises some known results related to identifiability of special types of mixtures and then discusses the general problem of identifiability in terms of mappings. Some new results follow for mappings with special features.

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.

Estimators which have locally uniform expansions are shown in this paper to be asymptotically equivalent to M-estimators. The M-functionals corresponding to these M-estimators are seen to be locally uniformly Fréchet differentiable. Other conditions for M-functionals to be locally uniformly Fréchet differentiable are given. An example of a commonly used estimator which is robust against outliers is given to illustrate that the locally uniform expansion need not be valid.