How is the PDF of a Non One-to-One Transformation Derived for n Dimensions?

[robbins]

This is not homework. Case I is mostly for background. The real questions are in Case II.

Case I (one dimension):
a. Suppose X is a continuous r.v. with pdf f_X(x), y = g(x) is one-to-one, and the inverse x = g^-1(y) exists. Then the pdf of Y = g(X) is found by
$$f_Y(y) = f_X(g^{-1}(y) | (g^{-1})'(y) |$$.
All of the above is well-known (http://en.wikipedia.org/wiki/Density_function" ).

b. Now suppose the transformation g is not one-to-one. Denote the real roots of y = g(x) by x_k, i.e., y = g(x₁) = ... = g(x_k). Then the pdf of Y = g(X) is
$$f_Y(y) = \sum_k = \frac{f_X(x_k)}{|g'(x_k)|}$$.
Again, this is (relatively) well-known.

Case II (n dimensions):
a. The case for n functions of n random variables where g is one-to-one is well-known (see link above):
$$f_Y(y) = f_X(g^{-1}(y))| J_{g^{-1}}(g(x)) |$$.
b. What about the case for n functions of n random variables where g is not one-to-one? Is there an analogous result to that in Case Ib? Can someone provide a reference?
c. What about the case for Y = g(X₁, x₂, ..., x_n) where g : Rⁿ -> R? [Note: Though g might still be one-to-one, we suppose that it is not.] The approach to Case IIc with which I am familiar is to construct an additional n-1 functions (often just using the identity function, but sometimes more effort is needed to choose functions so the Jacobian is nonzero), and then use the known method of Case IIa. Is there an approach analogous to that in Case Ib or Case IIb? Can someone provide a reference?

Thanks for any insights you can provide.

 

[winterfors]

You can write a symbolic solution to the PDF of any function $$Z=f(X)$$ of a stochastic variable $$X$$ as
$$p(z) = \int {p({ {x}})\delta \left( {z - f \left({x} \right)} \right)d{ {x}}}$$

where $$\delta(z)$$ is Dirac's delta function.

This works in any number of dimensions.

For theory, see e.g.
http://winterfors.wordpress.com/2008/02/21/mapping-of-a-probability-density/
or for some applications:
http://mathworld.wolfram.com/ProbabilityDensityFunction.html