I am self learning statistics. Right now I am learning about the multivariate change-of-variables technique, and now, I am on to the case where the transformation might not be one-to-one. A quick explanation, of what I think the method is for handling these cases:
That is, say we have random variables $X_1, X_2$ which have the support $-1<X_1, X_2 < 1$. They also have a uniform distribution over this bound. Then say we transform our variables into some $Y_1 = X_1^2$ and $Y_2 = X_2^2$.
Let's call the transformations that bring $X_1$ to $Y_1$ and $X_2$ to $Y_2$, $u_1(X_1)$ and $u_2(X_2)$ respectively. Thus, we can confidently say that $u_1, u_2$ is not one-to-one.
Also important to note are the inverses, $u_1^{-1}$ and $u_2^{-1}$, which are $\sqrt{Y_1}$ and $\sqrt{Y_2}$ respectively.
The support area $S$ before the transformation bounds from -1 to 1, while the transformed space $T$ goes from $0$ to $1$. Every point in $T$, except for (0, 0), has four corresponding points in $S$. Those are:
- $(x_1, x_2)$ such that $x_1$ = $\sqrt{y_1}$ and $x_2$ = $\sqrt{y_2}$
- $(x_1, x_2)$ such that $x_1$ = $-\sqrt{y_1}$ and $x_2$ = $\sqrt{y_2}$
- $(x_1, x_2)$ such that $x_1$ = $\sqrt{y_1}$ and $x_2$ = $-\sqrt{y_2}$
- $(x_1, x_2)$ such that $x_1$ = $-\sqrt{y_1}$ and $x_2$ = $-\sqrt{y_2}$
What we need to do is split $S$ into mutually disjoint subspaces $A_1, A_2, ... A_k$ so that the transformation from each of the $A_1, A_2, ... A_k$ to $T$ is one-to-one. Then, we can independently transform each of these subspaces and sum them up to get the effect of transforming all of $S$ to $T$.
The amount of mutually disjoint subspaces you create depends on how many corresponding points in $S$ each point in $T$ has. Here it has four, so $k =4$.
So, we have to split up $S$ into $A_1, A_2, A_3, A_4$ in a way such that the transformations from each A to S become one-to-one transformations.
Thus, the ending pdf will be something like
$$f(y_1, y_2) = \sum^4_{i=1} f_{x_1, x_2}(u_1^{-1}(y_1, y_2), u_1^{-1}(y_1, y_2))|J_i|$$
This is where I get shakey, though. What does this specifically expand to in this example? My guess is that it expands to something like this:
$$f(y_1, y_2) = f_{x_1, x_2}(\sqrt{y_1}, \sqrt{y_2})|J_1| + f_{x_1, x_2}(-\sqrt{y_1}, \sqrt{y_2})|J_2| + f_{x_1, x_2}(\sqrt{y_1}, -\sqrt{y_2})|J_3| + f_{x_1, x_2}(-\sqrt{y_1}, -\sqrt{y_2})|J_4|$$
Where each of the jacobians is the jacobian of each iteration's inverse function version, for example, $J_1$ looks like
$$J_1 = \begin{bmatrix} \frac{\partial x_1}{\partial y_1} & \frac{\partial x_1}{\partial y_2} \\ \frac{\partial x_2}{\partial y_1} & \frac{\partial x_2}{\partial y_2} \end{bmatrix} =\begin{bmatrix} \frac{\partial }{\partial y_1} \sqrt{y_1}& \frac{\partial }{\partial y_2} \sqrt{y_1} \\ \frac{\partial}{\partial y_1} \sqrt{y_2} & \frac{\partial }{\partial y_2} \sqrt{y_2} \end{bmatrix}=\begin{bmatrix} \frac{1}{2\sqrt{y_1}}& 0 \\ 0 & \frac{1}{2\sqrt{y_2}} \end{bmatrix}$$
And then we could calculate $$|J_1|=\frac{1}{4\sqrt{y_1y_2}}$$
The problem is I don't really know how to complete this explanation. How do we find these four $A_1, A_2, A_3, A_4$ subspaces?
Second question:
I'm specifically confused at this example in the book I'm reading, Introduction to Mathematical Statistics 8th edition by Hogg, Allen & Craig. The example states:
I understand all the stuff there - there are four different corresponding points in S for every point in T.
I don't understand how they get that value for the first jacobian, though. In my calculations, I do
$$J_1 = \begin{bmatrix} \frac{\partial x_1}{\partial y_1} & \frac{\partial x_1}{\partial y_2}\\ \frac{\partial x_2}{\partial y_1} & \frac{\partial x_2}{\partial y_2} \end{bmatrix} = \begin{bmatrix} \frac{\partial} {\partial y_1}\sqrt{y_1y_2} & \frac{\partial }{\partial y_2} \sqrt{y_1y_2}\\ \frac{\partial}{\partial y_1} \sqrt{y_1(1-y_2)} & \frac{\partial }{\partial y_2}\sqrt{y_1(1-y_2)} \end{bmatrix} = \begin{bmatrix} \frac{y_2}{2\sqrt{y_1y_2}} & \frac{y_1}{2\sqrt{y_1y_2}}\\ \frac{(1-y_2)}{2\sqrt{y(1-y_2)}} & -\frac{y_1}{2\sqrt{y_1(1-y_2)}} \end{bmatrix}$$
In summary: I would be so grateful if someone could complete my understanding on multivariable change-of-variables when we have a non-one-to-one function in general, and as well as this, make the example from the textbook make sense?
Thanks, A