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This question gives answer to the mean and variance of the zero inflated Poisson distribution, but is there an equation for the median or the percentile?

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Somewhat surprisingly, it seems like we don't have anything on quantiles for zero-inflated distributions, so I'll answer the more general question.

Assume you have a zero-inflated discrete distribution, i.e., your observation is zero with probability $\pi$ and comes from some other specified "vanilla" discrete distribution with probability $1-\pi$, so the probability mass function is

$$ f_\pi(y) = \begin{cases} \pi+(1-\pi)f_0(0), & \text{if }y=0 \\ (1-\pi)f_0(y), & \text{if }y=1,2.... \end{cases} $$

where $f_0$ denotes the PMF of the discrete "vanilla" distribution.

Consider a percentage $0\leq p<1$. Then the $p$-th quantile of the zero-inflated distribution is

$$q_\pi(p) = \begin{cases} 0, & \text{if } p\leq \pi+(1-\pi)f_0(0) \\ q_0\big(\frac{p-\pi}{1-\pi}\big), & \text{if } p> \pi+(1-\pi)f_0(0), \end{cases} $$ where $q_0(p)$ denotes the $p$-th quantile of the "vanilla" distribution.


In your specific case, the "vanilla" distribution is a Poisson distribution with parameter $\lambda$. There does not seem to be a closed form for the quantile of a "vanilla" Poisson distribution (R's ?qpois uses a search), so I would not expect there to be one for the more complicated zero-inflated case, either.


To see this, note first that if your target percentage $p$ fulfills $p\leq \pi+(1-\pi)f_0(0)$, then your percentile $q_\pi(p)$ is obviously zero.

So let us assume that $p>\pi+(1-\pi)f_0(0)$. We just follow the definition of a quantile:

$$ \begin{align} q_\pi(p) = & \min\Big\{y\,|\,\sum_{x=0}^y f_\pi(y)\geq p\Big\} \\ = & \min\Big\{y\,|\,\pi+(1-\pi)f_0(0)+ (1-\pi)\sum_{x=1}^yf_0(x)\geq p\Big\} \\ = & \min\Big\{y\,|\,\pi+(1-\pi)\sum_{x=0}^yf_0(x)\geq p\Big\} \\ = & \min\Big\{y\,|\,\sum_{x=0}^yf_0(x)\geq \frac{p-\pi}{1-\pi}\Big\} \\ = & q_0\big(\frac{p-\pi}{1-\pi}\big), \end{align}$$ as claimed.

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  • $\begingroup$ Ok this is really helpful, but could you walk through the calculation with a specific example? $\endgroup$
    – Daniel
    Commented Jun 22, 2021 at 15:28

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