Some new results on convolutions of heterogeneous gamma random variables

Convolutions of independent random variables often arise in a natural way in many applied areas. In this paper, we study various stochastic orderings of convolutions of heterogeneous gamma random variables in terms of the majorization order [p-larger order, reciprocal majorization order] of parameter vectors and the likelihood ratio order [dispersive order, hazard rate order, star order, right spread order, mean residual life order] between convolutions of two heterogeneous gamma sets of variables wherein they have both differing scale parameters and differing shape parameters. The results established in this paper strengthen and generalize those known in the literature.     

Stochastic comparisons in multivariate mixed model of proportional reversed hazard rate with applications

This paper studies the multivariate mixed proportional reversed hazard rate model having dependent mixing variables. Stochastic comparison as well as aging properties in this model are investigated, and stochastic monotone properties of the population vector with respect to the mixing vector are also discussed. Moreover, MTP₂ dependence among the mixing vectors is proved to imply the increasingness of the reversed hazard rate with respect to the baseline one. Finally, some interesting applications are presented as well.     

On the conjecture of Kochar and Korwar

In this article, we partially solve a conjecture by Kochar and Korwar (1996) [9] in relation to the normalized spacings of the order statistics of a sample of independent exponential random variables with different scale parameters. In the case of a sample of size n=3, they proved the ordering of the normalized spacings and conjectured that result holds for all n. We prove this conjecture for n=4 for both spacings and normalized spacings and generalize some results to n>4.     

Stochastic comparisons for time transformed exponential models

Different sufficient conditions for stochastic comparisons between random vectors have been described in the literature. In particular, conditions for the comparison of random vectors having the same copula, i.e., the same dependence structure, may be found in Müller and Scarsini (2001). Here we provide conditions for the comparison, in the usual stochastic order sense and in other weaker stochastic orders, of two time transformed exponential bivariate lifetimes having different copulas. Some examples of applications are provided too.     

Some new results on multivariate dispersive ordering of generalized order statistics

Let be generalized order statistics based on a continuous distribution function F with parameters k and (m₁,…,m_n−1). Chen and Hu (2007) [8] investigated the sufficient conditions on F and on the parameters k and m_i’s such that , where , and is the Shaked-Shanthikumar multivariate dispersive order. Since the order does not possess the closure property under marginalization, one may naturally wonder whether the corresponding multivariate margins of the above random vectors are also ordered in the order . This is answered affirmatively in this paper. Some comparison results for generalized order statistics from two samples are presented. Potential applications are also mentioned.     

Comparison of two repairable systems

Any repairable system improves (deteriorates) with time if the interarrival times of failure tend to get larger (smaller) in some sense. In this paper we consider two such repairable systems, and their performance in terms of several partial orderings of their respective interarrival times of failure are compared. The comparison of two systems’ improvement/deterioration under minimal repair policy has been characterized in terms of s-FR orders and also in terms of their shifted and dispersive versions. These results generalize some of the existing results in the literature and also provide some new results in this direction.     

On computing signatures of coherent systems

It is difficult to compute the signature of a coherent system with a large number of components. This paper derives two basic formulas for computing the signature of a system which can be decomposed into two subsystems (modules). As an immediate application, we obtain the formula for computing the signature of systemwise redundancy in terms of the signatures of the original system and the backup one. The formula for computing the signature of a componentwise redundancy system is also derived. Some examples are given to illustrate the power of the main results.     

Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (_Xn,_ηn)$(X_n,{\eta }_n)$ on _Z+×S${\mathbb{Z}}_{+}\times S$, where _Z+${\mathbb{Z}}_{+}$ is the non-negative integers and S$S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of _Xn$X_n$, and that, roughly speaking, _ηn${\eta }_n$ is close to being Markov when _Xn$X_n$ is large. This departure from much of the literature, which assumes that _ηn${\eta }_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for _Xn$X_n$ given _ηn${\eta }_n$. We give a recurrence classification in terms of increment moment parameters for _Xn$X_n$ and the stationary distribution for the large- X$X$ limit of _ηn${\eta }_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between _Xn$X_n$ (rescaled) and _ηn${\eta }_n$. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on _Z+${\mathbb{Z}}_{+}$ (the case where S$S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where _ηn${\eta }_n$ tracks an internal state of the system.     

Stationarity of multivariate particle systems

A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in R^d${\mathbb{R}}^d$ and in some cases provide a full characterisation of the stationarity property. In particular, a full characterisation of stationary multivariate Brown–Resnick processes is given.     

Conditional residual lifetimes of coherent systems

In this paper, we derive mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n$n$ independent and identically distributed (i.i.d.) components under the condition that at least j$j$ and at most k−1$k-1$ (j<k$j<k$) components have failed by time t$t$. Based on these mixture representations, we then discuss stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components.     

Multivariate linear recursions with Markov-dependent coefficients

We study a linear recursion with random Markov-dependent coefficients. In a “regular variation in, regular variation out” setup we show that its stationary solution has a multivariate regularly varying distribution. This extends results previously established for i.i.d. coefficients.     

Stationary-excess operator and convex stochastic orders

The present paper aims to point out how the stationary-excess operator and its iterates transform s-convex stochastic orders and the associated moment spaces. This allows us to propose a new unified method on constructing s-convex extrema for distributions that are known to be t-monotone. Both discrete and continuous cases are investigated. Several extremal distributions under monotonicity conditions are derived. They are illustrated with some applications in insurance.     

On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables

Some Kolmogorov probability inequalities for quadratic forms and weighted quadratic forms of negative superadditive dependent (NSD) uniformly bounded random variables are provided. Using these inequalities, some complete convergence of randomized quadratic forms under some suitable conditions are evaluated. Moreover, various examples are presented in which the given conditions of our results are satisfied.     

Ewens sampling formulae with and without selection

We shall first consider the random Dirichlet partitioning of the interval into n   fragments at temperature θ>0.$\theta >0.$ Using calculus for Dirichlet integrals, pre-asymptotic versions of the Ewens sampling formulae from finite Dirichlet partitions follow up. From these preliminaries, straightforward proofs of the usual sampling formulae from random proportions with Poisson–Dirichlet (PD)(γ)$(\mathrm{PD})(\gamma )$ distribution can be obtained, while considering the Kingman limit n↗∞$n\nearrow \infty$, θ↘0$\theta \searrow 0$, with nθ=γ>0$n\theta =\gamma >0$.     

Logarithmic speeds for one-dimensional perturbed random walks in random environments

We study the random walk in a random environment on Z⁺={0,1,2,…}${\mathbb{Z}}^{+}=\{0,1,2,\dots \}$, where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) a random walk in a random environment perturbed from Sinai’s regime; (ii) a simple random walk with a random perturbation. We give almost sure results on how far the random walker is from the origin, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order (logt)^β${(logt)}^{\beta }$, for β∈(1,∞)$\beta \in (1,\infty )$, depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution.     

Multivariate copulas, quasi-copulas and lattices

We investigate some properties of the partially ordered sets of multivariate copulas and quasi-copulas. Whereas the set of bivariate quasi-copulas is a complete lattice, which is order-isomorphic to the Dedekind-MacNeille completion of the set of bivariate copulas, we show that this is not the case in higher dimensions.     

Exponential families of mixed Poisson distributions

If I=(I₁,…,I_d) is a random variable on [0,∞)^d with distribution μ(dλ₁,…,dλ_d), the mixed Poisson distribution MP(μ) on N^d is the distribution of (N₁(I₁),…,N_d(I_d)) where N₁,…,N_d are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)^d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v₀+v₁Y₁+?+v_qY_q for some q?d, for some vectors v₀,…,v_q of [0,∞)^d with disjoint supports and for independent standard real gamma random variables Y₁,…,Y_q.     

Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.