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

We derive a multidimensional Saddlepoint solution as an alternative to (7.4). 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]

$\displaystyle p({\bf u}) = (2\pi)^{-M/2} \; \frac{ \exp\left\{ c_z(\hat{\mbox{\... {\bf u}\right\} } {\sqrt{{\rm det} (C_z(\hat{\mbox{\boldmath$\lambda$}}))} }$ (7.7)

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

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

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($$ \lambda$$ )$ :

$\displaystyle C_z($$ \lambda$$\displaystyle )_{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

$\displaystyle {\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.

Baggenstoss 2017-05-19