How to scale a generalized gamma distribution?

Question

Is the following derivation for scaling a Generalized Gamma distribution correct?

Given $X\sim GG(x;a,d,p)$, with $x\ge 0$, $a,d,p > 0$ and pdf

$$f_x(x;a,d,p) = \frac{px^{d-1}exp\Big(-(x/a)^p\Big)}{a^d\Gamma(d/p)}$$

Let $Y=g(X)=kX$ and, consequently, $X=g^{-1}(Y)=Y/k$, and the Jacobian $dX/dY=1/k$. Then, the transformation formula holds:

$$f_y(y)=f_x(g^{-1}(y))\left|\frac{dx}{dy}\right|$$

Hence we have

$$f_y(y;ka,d,p) = \frac{py^{d-1}exp\Big(-(\frac{x}{ka})^p\Big)}{(ka)^d\Gamma(d/p)}$$

Answer 1

The general approach is correct but you have mistakenly written $\exp(-(\frac{x}{ka})^p)$ rather than $\exp(-(\frac{y}{ka})^p)$. The conclusion is that if $X$ is generalized with parameters $a$, $d$, $p$ (which are scale, shape, and "power," respectively), then $Y = kX$ is generalized with scale $ka$, with the same shape and power parameters. This makes sense since this is consistent with the notion of a scale parameter.