Sample distribution theory using Coarea Formula
Luigi Negro · Communication in Statistics- Theory and Methods · 2022
Let (Ω,Σ,p) be a probability measure space and let X:Ω→Rk be a (vector valued) random variable. We suppose that the probability pX induced by X is absolutely continuous with respect to the Lebesgue measure on Rk and set fX as its density function. Let ϕ:Rk→Rn be a C1-map and let us consider the new random variable Y=ϕ(X):Ω→Rn. Setting m:=max{rank (Jϕ(x)):x∈Rk}, we prove that the probability pY induced by Y has a density function fY with respect to the Hausdorff measure Hm on ϕ(Rk) which satisfies fY(y)=∫ϕ−1(y)fX(x)1Jmϕ(x) dHk−m(x), for Hm−a.e. y∈ϕ(Rk). Here Jmϕ is the m-dimensional Jacobian of ϕ. When Jϕ has maximum rank we allow the map ϕ to be only locally Lipschitz. We also consider the case of X having probability concentrated on some m-dimensional sub-manifold E⊆Rk and provide, besides, several examples including algebra of random variables, order statistics, degenerate normal distributions, Chi-squared and “Student's t” distributions.