Table 3.

The alternative hypothesis H_A:p_g>e^−λ is robust to different model hypotheses

Underlying distribution	p under H0	p under HA
Mixture of Poisson	$e^{-\lambda }=e^{-\sum _k{\pi }_k{\lambda }_k}$	$\sum _k{\pi }_k{e}^{-{\lambda }_k}$
Negative binomial	e−λ	${\left(\frac{r}{r+\lambda }\right)}^r$
Zero-inflated negative binomial	e−λ	${\left(\frac{r}{r+\lambda }\right)}^r+{\pi }_0$

In the first row, the right column is larger than the left column due to Jensen’s inequality. For negative binomial, the dispersion parameter r is constructed so that the variance is $\frac{{\lambda }^2}{r}+\lambda$, so that Poisson is a special case of negative binomial with r=∞. The zero-inflated negative binomial distribution is parameterized as π₀δ₀+(1−π₀) NB(λ,r)