Quantile function for a mixture model

Question

Suppose a mixture model, having the following distribution function:

\begin{equation} f(x) = \sum_{k = 0}^K \pi_k f_k(x) \end{equation}

where $f_k(x)$ is a Poisson distribution with parameter $\lambda_k$.

Is it possible to have an analytical expression of the quantile function of the mixture model using the quantile function of the individual Poisson component (or the repartition function or any usual function ?) ?

Answer 1

While the quantile function of a Gaussian mixture distribution sounds impossible to represent in a closed form, the cdf of the mixture is available as$$\sum_{i=1}^k \pi_i F_i(\cdot)$$which means it is possible to check whether or not one value is an $\alpha$ quantile and hence to apply numerical divide-and-conquer strategies. An example is provided in this other CV question.