# Zero-inflated gamma - how to write down the cdf?

My goal is building a predictive model to give probabilistic forecasts.

My response variable has lots of zeros but otherwise looks close to a gamma.

I fit the whole dataset using some classification techniques and have probabilities for zero vs. nonzero for each record.

Now I run a gamma GLM on just the records with positive response.

I think in this case I have a mixture of a degenerate distribution at 0 and a gamma. How can I derive the cdf of a prediction?

I assume from the way your question is phrased you're conditioning on the estimated parameters (if you're not - i.e. if you're incorporating the impact of the uncertainty in the esitmation of the parameters, it's somewhat more complicated).

As you say, it's just a finite mixture of a degenerate distribution at 0 and a gamma.

So if your degenerate has distribution $H_0(x)$ and your gamma has cdf $G(x)$, and they occur with probabilities $p$ and $1-p$ respectively, then

$$F(x) = pH_0(x)+(1-p)G(x)$$

So

$$F(x)= \begin{cases} 0 & \; x \leq 0 \\ p+(1-p)G(x) & \; x \gt 0 \end{cases}$$

where $G(x) = \frac{\gamma\left(\alpha, \frac{x}{\beta}\right)}{\Gamma(\alpha)}$ (for the scale-shape form of the gamma, where $\gamma$ is the lower incomplete gamma function).