Implement the generalized normal distribution

Description

Implement Generalized normal distribution https://en.wikipedia.org/wiki/Generalized_normal_distribution

Why this is useful?

It generalizes the normal (for p=2) and double exponential (Laplace) distribution (for p=1) and in the limit case also the uniform (p → ∞) for the shape parameter p.

These correspond to the $L_p$ norms when used for regularization and in other cases.

For examples:

y ~ normal(X*beta, sigma);

produces the criterion minimize $L_2$ norm of (X*beta-y) (a.k.a. least squares). But then

y ~ double_exponential(X*beta, sigma);

produces the criterion minimize $L_1$ norm of (X*beta-y). (a.k.a. least absolute deviations)

Similarly, in the Bayesian interpretation of ridge and LASSO,

beta ~ normal(0, lambda);
y ~ normal(X*beta, sigma);

produces the $L_2$ regularized ridge and

beta ~ double_exponential(0, lambda);
y ~ normal(X*beta, sigma);

produces the $L_1$ regularized LASSO.

Using the Generalized normal distribution would allow to conveniently use an arbitrary $L_p$ norm for the optimization criterion, or to even find the suitable value of p when used as a parameter.

Start by reviewing the existing normal and double_exponential entry points and the linked Wikipedia definition. Establish the required parameterization, supported inputs, derivatives, edge cases, and test coverage for the new distribution. Done means the generalized distribution is implemented consistently with the library and its behavior is verified across relevant cases.