Novel goodness-of-fit tests for binomial count time series

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

“…Utilizing the binomial Stein identity (2), Aleksandrov et al. (2021) proposed a binomial GoF‐test based on the statistic $$$$\begin{equation} \hat{\mbox{T}}\!_{f;\,{\rm Bin}} =\frac{{\left(N-\overline{X}\right)}\ \overline{X\,f(X)}}{\, \overline{X}\ \overline{(N-X)\,f(X+1)}}, \end{equation}$$$$where $$$$\begin{equation} \mbox{T}\!_{f;\,{\rm Bin}}=\frac{{\left(N-E[X]\right)}\,E{\left[X\,f(X)\right]}}{E[X]\,E{\left[(N-X)\,f(X+1)\right]}} \quad \text{equals\nobreakspace 1 under the Bin-null}. \end{equation}$$$$Note that the statistic $$\mbox{T}\!_{f;\,{\rm Bin}}$$ in (25) (as well as the asymptotics in Theorem 2) can again be expressed by using the moments …”

“…It covers the special case of $$\mbox{Bin}(N,\pi )$$‐counts with exponential weighting scheme $$f(x)=\exp (-x)$$ as considered by Aleksandrov et al. (2021).…”

“…The proof of Theorem 2 is provided in Appendix A.4. It covers the special case of Bin(𝑁, 𝜋)-counts with exponential weighting scheme 𝑓(𝑥) = exp(−𝑥) as considered by Aleksandrov et al (2021).…”

“…To derive the asymptotics of the Stein-type Bin-GoF statistic (24) for i.i.d. bounded counts with unspecified distribution, we extend the derivations of Aleksandrov et al (2021) (A.24)…”