Saddlepoint Approximation and First-Order Correction Term to the Joint Probability Density Function of M Quadratic and Linear Forms in K Gaussian Random Variables With Arbitrary Means and Covariances
NAVAL UNDERSEA WARFARE CENTER NEWPORT DIV RI
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Let w be a Kx1 Gaussian random vector with arbitrary Kx1 mean vector r and KxK covariance matrix R. The general quadratic and linear forms of interest are the M random scalars zm w Pm w pm w pm for m1 M, where KxK matrix Pm, Kx1 vector pm, and scalar qm contain arbitrary constants for m1M. The joint probability density function PDF of Mx1 random vector zZ1... ZM at an arbitrary point in M-dimensional space is desired. An exact expression for the joint moment generating function MGF of random vector z is derived. The inability analytic and numerical to perform the M-dimensional inverse Laplace transform back to the PDF domain requires use of the saddlepoint approximation SPA to obtain useful numerical values for the desired PDF of z. A first-order correction term to the SPA is also employed for more accuracy, which requires fourth-order partial derivatives of the joint cumulate generating function CGF. Derivation of the fourth-order partial derivatives of the CGF involves some interesting and useful matrix manipulations which are fully developed. Two MATLAB programs for the entire SPA procedure with correction term are presented in this report.
- Statistics and Probability