Fraser A Daly

Stein's method and stochastic orderings

A stochastic ordering approach is applied with Steins method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

On magic factors in Stein’s method for compound Poisson approximation

One major obstacle in applications of Steins method for compound Poisson approximation is the availability of so-called magic factors (bounds on the solution of the Stein equation) with favourable dependence on the parameters of the approximating compound Poisson random variable. In general, the best such bounds have an exponential dependence on these parameters, though in certain situations better bounds are available. In this paper, we extend the region for which well-behaved magic factors are available for compound Poisson approximation in the Kolmogorov metric, allowing useful compound Poisson approximation theorems to be established in some regimes where they were previously unavailable. To illustrate the advantages offered by these new bounds, we consider applications to runs, reliability systems, Poisson mixtures and sums of independent random variables.

Compound geometric approximation under a failure rate constraint

We consider compound geometric approximation for a nonnegative, integer-valued random variable

Upper Bounds for Stein-Type Operators

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Compound Poisson approximation with association or negative association via Stein's method

. The bound we give is straightforward but relies on having a lower bound on the failure rate of

Stein's method for compound geometric approximation

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Negative dependence and stochastic orderings

. Applications are presented to M/G/1 queuing systems, for which we state explicit bounds in approximations for the number of customers in the system and the number of customers served during a busy period. Other applications are given to birth-death processes and Poisson processes.