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A stochastic ordering approach is applied with Stein's 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 approximation of a sum of dependent Bernoulli random variables, for example k-runs in i.i.d. Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

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.

The Conway-Maxwell-Poisson (CMP) distribution is a natural two-parameter generalisation of the Poisson distribution which has received some attention in the statistics literature in recent years by offering flexible generalisations of some well-known models. In this work, we begin by establishing some properties of both the CMP distribution and an analogous generalisation of the binomial distribution, which we refer to as the CMB distribution. We also consider some convergence results and approximations, including a bound on the total variation distance between a CMB distribution and the corresponding CMP limit. A two-parameter generalisation of the Poisson distribution was introduced by Conway and Maxwell [10] as the stationary number of occupants of a queuing system with state dependent service or arrival rates. This distribution has since become known as the Conway-Maxwell-Poisson (CMP) distribution. Beginning with the work of Boatwright, Borle and Kadane [7] and Shmueli et al. [31], the CMP distribution has received recent attention in the statistics literature on account of the flexibility it offers in statistical models. For example, the CMP distribution can model data which is either under-or over-dispersed relative to the Poisson distribution. This property is exploited by Sellers and Shmueli [29], who use the CMP distribution to generalise the Poisson and logistic regression models. Kadane et al. [24] considered the use of the CMP distribution in Bayesian analysis, and Wu, Holan and Wilkie [34] use the CMP distribution as part of a

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