## The Nonlinear Case

If we denote G (x) = Ax + b, G-i(y) = A-i( y – b), then the result of Theorem 4.3 reads h(y) = f (G-i(y))|det(9G-i(y)/9y)|. This suggests that Theorem 4.3 can be generalized as follows:

Theorem 4.4: Let X be k-variate, absolutely continuously distributed with joint density f(x),x = (xi,…,xk)T, and let Y = G(X), where G(x) = (gi(x),…, gk(x))T is a one-to-one mapping with inverse mapping x = G-i(y) = (gj(y),…,gl(y))T whose components are differentiable in the components ofy = (yi;yk)T. Let J(y) = dx/dy = dG-i(y)/dy, that is, J(y) is the matrix with i, j’s element dgf(y)/dyj, which is called the Jacobian...

Read More