Goodness‐of‐fit tests for Poisson count time series based on the Stein–Chen identity

Abstract

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.

Change Detection by Monitoring Residuals from Time Series Models

“…The SC bound does not seem to be commonly used; however, related references are available [5][6][7][8][9][10]. For example, Ref.…”

“…[7] applies the SC method to calculate coincidence probabilities. References [8,9] apply the SC method in different time series contexts than considered here. To simplify the calculation of bivariate Poisson moments, Ref.…”

“…The probability p is the probability that the original time series X exceeds a threshold, and the I notation denotes an indicator or binary variable. As an aside, the SC method can also be applied if p is not constant over time, but the independence assumption is difficult to avoid [5][6][7][8][9][10]. Any type of time series model [1,2] can be fit, and then the resulting residuals become the original series that is thresholded to convert to binary; therefore, the application is quite general.…”

Optimal Stein‐type goodness‐of‐fit tests for count data

“…In Aleksandrov et al. (2022), it is suggested to use the Poisson identity (1) with an appropriate choice of f for developing a moment‐based GoF‐test for the Poisson distribution (note that different types of GoF‐test, but also being related to a Stein characterization, are derived by Yang et al. (2018) and Betsch et al.…”

“…In their simulation experiments, Aleksandrov et al. (2022) observed that the empirical power of the tests for different alternatives strongly depends on the choice of f . For example, for an NB‐alternative, the choice $$f(x)=|x-1|^a$$ with a close to 1 leads to a good power, whereas $$f(x)=\exp (-x)$$ appears to be a good choice for a zero‐inflated Poisson (ZIP) alternative.…”