Estimating the parameters of a sum of a Gaussian and an $alpha$-stable random variable

Let’s assume I have a set of samples of a random variable

$$
X = Y + Z >,
$$

where $Y$ is Gaussian (with a mean of zero and variance $sigma^2$) and $Z$ has a symmetric $alpha$-stable distribution with index of stability $alpha$ and scale parameter $c$. I know $alpha$, but I don’t know $sigma$ and $c$. How do I go about estimating them?

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