Lets say that I have variable $X$ which follows a generalized normal distribution with parameters $(\beta, \mu, \sigma)$ and I wish to change it so that it has parameters $(\beta, \tilde{\mu}, \tilde{\sigma})$

I thought that I could define

$$ Y = \tilde{\sigma}\frac{X- \mu}{\sigma} + \tilde{\mu} $$


$$ \mathbb{E}[Y] = \mathbb{E} \left[ \tilde{\sigma}\frac{X- \mu}{\sigma} + \tilde{\mu} \right] = \frac{\tilde{\sigma}}{\sigma} \mathbb{E}[X-\mu] + \tilde{\mu} = \tilde{\mu} \\ \mathbb{V}[Y] = \mathbb{V}\left[ \tilde{\sigma}\frac{X- \mu}{\sigma} + \tilde{\mu} \right] = \frac{\tilde{\sigma}^2}{\sigma ^2} \mathbb{V}[X- \mu] = \tilde{\sigma}^2 $$ But after I implemented in python and tried, this doesnt seem to be correct. I get the desired beta, and mean $\beta, \tilde{\mu}$. But the standard deviation $\tilde{\sigma}$ is way off.

What is the correct approach?


In the generalized normal distribution, we have a pdf of the form:

$p_X(x) \sim \exp\left(-\left(\frac{|x-\mu|}{\alpha}\right)^\beta\right)$

If $\beta=2$ then this is exactly the normal distribution and $\alpha^2$ corresponds to $2\sigma^2$. The transform you give is also correct in general, except that $\sigma$ should be changed to $\alpha$:

$Y = \tilde \alpha \left( \frac{X-\mu}{\alpha} \right) + \tilde \mu$.

I suspect you are confused about the difference between $\alpha$ and the standard deviation $\sigma$. They are closely related, but the ratio between them changes with $\beta$.

In general the variance will be (see https://en.wikipedia.org/wiki/Generalized_normal_distribution)

$\sigma^2 = \frac{\alpha^2 \Gamma(3/\beta)}{\Gamma(1/\beta)}$

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