Studies stochastic models of queueing, reliability, inventory, and sequencing in which random influences are considered. One stochastic mode--rl is approximated by another that is simpler in structure or about which simpler assumptions can be made. After general results on comparison properties of random variables and stochastic processes are given, the properties are illustrated by application to various queueing models and questions in experimental design, renewal and reliability theory, PERT networks and branching processes.

https://link.springer.com/content/pdf/10.1057%2Fjors.1984.224.pdf

Comparison Methods for Queues and Other Stochastic Models

A connection between supermodular ordering and positive/negative association

In this article we present a sequence of random variables with weighted tail distribution functions, constructed based on the relevation transform. For this sequence, we prove several recursive formulas and connections to the residual entropy through the unifying framework of the Dickson-Hipp operator. We also give some numerical examples to evaluate our results.

Some extensions of the residual lifetime and its connection to the cumulative residual entropy

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

Generalized Increasing Convex and Directionally Convex Orders

Stochastic ordering of random variables may be defined by the relative convexity of the tail functions. This has been extended to higher order stochastic orderings, by iteratively reassigning tail-weights. The actual verification of stochastic orderings is not simple, as this depends on inverting distribution functions for which there may be no explicit expression. The iterative definition of distributions, of course, contributes to make that verification even harder. We have a look at the stochastic ordering, introducing a method that allows for explicit usage, applying it to the Gamma and Weibull distributions, giving a complete description of the order of relations within each of these families.

/pdf/iterated-failure-rate-monotonicity-and-ordering-relations-3o4xdruhqh.pdf

Iterated failure rate monotonicity and ordering relations within gamma and weibull distributions

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/negative dependence concepts and convex ordering are also presented. These results lead to approximations and bounds for the distributions of sums of positive/negative dependent random variables. Applications and extensions of the main results pertaining to compound Poisson, normal and exponential approximation are provided as well.

/pdf/on-the-distance-between-convex-ordered-random-variables-with-3n0guuhmp3.pdf

On the distance between convex-ordered random variables, with applications

Let X1, X2, …, Xn be a sequence of independent or locally dependent random variables taking values in ℤ+. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum ∑i=1nXi and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This “smoothness factor” is of order O(σ−2), according to a heuristic argument, where σ2 denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.

/pdf/a-new-method-for-obtaining-sharp-compound-poisson-2npbgoy4f8.pdf

A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables

Let $X_1,X_2,...,X_n$ be a sequence of independent or locally dependent random variables taking values in $\mathbb{Z}_+$. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum $\sum_{i=1}^nX_i$ and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This "smoothness factor" is of order $\mathrm{O}(\sigma ^{-2})$, according to a heuristic argument, where $\sigma ^2$ denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.

Eutichia Vaggelatou

Papers

On the distance between convex-ordered random variables, with applications

A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables

A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables