A good check on the analysis is to compare the two methods just presented in Sections 7.2 and 7.1.2.

We tried the case of $ N=100$, $ M=3$, $ t_1=3$, $ t_2=7$. Samples of input data vector $ {\bf x}$ were generated using $ N$ independent uniformly distributed RVs (in the range 0 to 1), then scaled by a random scale factor in the range 0 to 10. Note that the PDF used for data generation is not important because we are comparing two PDF approximations for the same input feature vector. The PDF of $ {\bf z}$ was computed using the methods of Sections 7.2 and 7.1.2. The results are shown in Figure 7.1. The two methods agreed very closely - within a maximum error of .059 in log space.

Figure 7.1: Comparison of Saddlepoint Approximation (Section 7.2) with integral solution (Section 7.1.2). Largest difference was .059
\includegraphics[width=4.5in,height=2.5in, clip]{ordcomp.eps}

Baggenstoss 2017-05-19