exponential power distribution

Question

I see two different definitions of the exponential power distribution:

http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf

$$f(x)= \exp(1-\exp(\lambda x^k)) \exp(\lambda x^k) \lambda k x^{k-1}$$

and from wikipedia,

https://en.wikipedia.org/wiki/Generalized_normal_distribution#:~:text=Known%20also%20as%20the%20exponential,intervals%20of%20the%20real%20line.

$$f(x)=\frac{\beta}{2\alpha\Gamma(1/\beta)} \; e^{-(|x-\mu|/\alpha)^\beta}$$

are they equivalent? which one is more common? why do they receive the same name?

Answer 1

1. They are most definitely not equivalent, simply as the first distribution is defined for $x > 0$ and the second is defined for other $x$ as well. Or more formally, the former always gives $0$ mass to negative numbers, whereas the latter always gives some positive mass there.

2. I have seen the second distribution more often.

3. Poor naming? The second distribution becomes normal for $\beta = 2$, whereas I do not see how the first is related to it. Instead, it is a reminiscent of extreme value distributions