SPA Solution for Order Statistics of Exponential RVs (module_exp_ord.m)

We derive a multidimensional Saddlepoint solution as an alternative to (8.4). We consider only the first case (exponential RVs). This will provide a means of validating the integral solution of the previous section.

Let $p_z({\bf z})$ be a PDF defined on the space ${\cal R}^M$. The saddlepoint approximation is given by [16]

$$p({\bf u}) = (2\pi)^{-M/2} \; \frac{ \exp\left\{ c_z(\hat{\mbox{\... ...me {\bf u}\right\} } {\sqrt{{\rm det} (C_z(\hat{\mbox{\boldmath$\lambda$}}))} }$$ (8.7)

where $c_z($$\lambda$$)$ is the joint cumulant generating function (CGF)

$$c_z($$$\lambda$$$) = \log g_z($$$\lambda$$$),$$

and $g_z($$\lambda$$)$ is the joint moment function of $p_z({\bf z})$, and $C_z($$\lambda$$)$ is the $M\times M$ Hessian matrix of $c_z($$\lambda$$)$ :

$$C_z($$$\lambda$$$)_{i,j} = \frac{\partial^2 c_z(\mbox{\boldmath $\lambda$})} {\partial \lambda_i \partial \lambda_j},$$

and $\hat{\mbox{\boldmath $\lambda$}}$ is the saddle point, the value of $\lambda$ where

$${\partial c_z(\mbox{\boldmath $\lambda$}) \over \partial \lambda_m} = u_m, \;\;\; 1\leq m \leq M.$$

Thus, to obtain the SPA, we need the joint CGF and its first and second-order partial derivatives.