# How to scale a generalized gamma distribution?

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

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

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.