Multivariate Transformations
- Statistics 4.3
Multivariate Transformations
A multivariate transformation is a function that maps a vector of random variables to another vector of random variables. Let $\mathbf{X} \sim f_{\mathbf{X}}(\mathbf{x})$ be a random vector at $\mathbb{R}^n$ and $\mathbf{Y} = T(\mathbf{X})$ be a transformation of $\mathbf{X}$. Here, $T: \mathbb{R}^n \to \mathbb{R}^n$ is one-to-one and differentiable. Then the distribution of $\mathbf{Y}$ is given by:
\[ f_{\mathbf{Y}}(\mathbf{y}) = f_{\mathbf{X}}\left(T^{-1}(\mathbf{y}) \right) \left| \det \pdv{T^{-1}(\mathbf{y})}{\mathbf{y}} \right| \]
where $\det \pdv{T^{-1}(\mathbf{y})}{\mathbf{y}}$ is the determinant of the Jacobian matrix of the inverse transformation $T^{-1}$.
Proof
\[ \begin{align*} P(\mathbf{Y} \in A) &= P(T(\mathbf{X}) \in A) = P(\mathbf{X} \in T^{-1}(A)) \nl &= \int_{T^{-1}(A)} f_{\mathbf{X}}(\mathbf{x}) \, \dd{\mathbf{x}} \nl &= \int_{A} f_{\mathbf{X}}\left(T^{-1}(\mathbf{y}) \right) \left| \det \pdv{T^{-1}(\mathbf{y})}{\mathbf{y}} \right| \, \dd{\mathbf{y}} \nl &= \int_{A} f_{\mathbf{Y}}(\mathbf{y}) \, \dd{\mathbf{y}} \end{align*} \]
If $T$ is not one-to-one, we can think of the partition of the domain into disjoint sets where $T$ is one-to-one. In this case, we can apply the transformation to each partition and sum the contributions. We can construct the process just as we did for the univariate case.
\[ f_{\mathbf{Y}}(\mathbf{y}) = \begin{cases} \dps \sum_{i} f_{\mathbf{X}}\left(T^{-1}_i(\mathbf{y}) \right) \left| \det \pdv{T^{-1}_i(\mathbf{y})}{\mathbf{y}} \right| \mathbf{1}_{T_i(\mathbf{X})} (\mathbf{y}) & ; \mathbf{y} \in \supp f_{\mathbf{Y}} \nl 0 & ; \mathbf{y} \notin \supp f_{\mathbf{Y}} \end{cases} \]
