An Extension of Plackett’s Differential Equation for the Multivariate Normal Density
Simeon M. Berman · SIAM Journal on Algebraic and Discrete Methods · 1987
Let $f({\bf x},{\bf y};B)$, with $x,y$ in $R^m $, and B a nonsingular real $m \times m$ matrix, be a function of the form \[ f = ( 2\pi )^{m/ 2} | \det B |^{1/2} \exp \left( - \frac{1}{2}x' B^{ - 1} y \right). \] It is shown that f satisfies a partial differential equation which represents a generalization of Plackett’s equation in the case where B is positive definite, that is, where f is a normal density in m dimensions.