multidimensional change of variables

Question

If $\mathbf{X}=(X_1,...,X_n)$ s.t $X_i$'s are iid r.vs with $X_i\sim Ga(\alpha_i,\beta_i)$. Let $S=\sum_{j=1}^n X_j$ and $R_i={X_i\over S}$ s.t $\mathbf{R}=(R_1,...,R_n)$. I have read that the density of $f_{\mathbf{R},S}(r_1,...,r_n,s)$ is $$f_{\mathbf{R},S}(r_1,...,r_n,s)=f_{\mathbf{X}}(r_1s,...,r_ns)s^{n-1}.$$ How did we make the variable change from $n$ to $n+1$ variables and should $|J|=s^{n}$?

Answer 1

$S$ should be $\displaystyle\sum_{j=1}^n X_j.$ The joint distribution of $(X_1,\dots,X_n,S)$ is the same as that of $(X_1,\dots,X_n)\mathbb{I}(S=\displaystyle\sum_{j=1}^n X_j).$ $$f_{X_1,\dots,X_n,S}(x_1,\dots,x_n,s)=\begin{cases}\displaystyle\prod_{i=1}^ng(x_i;\alpha_i,\beta_i)\hspace{1cm}\text{if }x_i>0,\,\,s=\displaystyle\sum_{i=1}^nx_i \\0\hspace{4cm}\text{otherwise,}\end{cases}$$ where $g(.;\alpha_i,\beta_i)$ is the pdf of the $i^{\text{th}}$ Gamma distribution. Now we use transformation of random variables to get the required distribution. Let $(R_1,\dots,R_n,S)=h(X_1,\dots,X_n,S).$ The inverse transformation should be $(X_1,\dots,X_n,S)=(R_1S,\dots,R_nS,S).$ Then the Jacobian is

$$\begin{bmatrix} s & 0 &\dots &0 &r_1 \\ 0& s & 0 &\dots &r_2 \\ 0&0 &s &\dots &r_3 \\ \vdots& \vdots & &\ddots &\vdots \\ 0& 0& \dots &0 &1 \end{bmatrix}$$

and so $|J|=s^{n}.$