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To test the null hypothesis of a Poisson marginal distribution, test statistics based on the Stein-Chen identity are proposed. For a wide class of Poisson count time series, the asymptotic distribution of different types of Stein-Chen statistics is derived, also if multiple statistics are jointly applied. The performance of the tests is analyzed with simulations, as well as the question which Stein-Chen functions should be used for which alternative. Illustrative data examples are presented, and possible extensions of the novel Stein-Chen approach are discussed as well.

After having fitted a model to a given count time series, one has to check the adequacy of this model fit. The (standardized) Pearson residuals, being easy to compute and interpret, are a popular diagnostic approach for this purpose. But which types of model inadequacy might be uncovered by which statistics based on the Pearson residuals? In view of being able to apply such statistics in practice, it is also crucial to ask for the properties of these statistics under model adequacy. We look for answers to these questions by means of a comprehensive simulation study, which considers diverse types of count time series models and inadequacy scenarios. We illustrate our findings with two real-data examples about strikes in the U.S., and about corporate insolvencies in the districts of Rhineland–Palatinate. We conclude with a theoretical discussion of Pearson residuals.

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