Probability Density Function (pdf_ord_rem_chisq.m)

Interestingly, $ p({\bf z}\vert H_0)$ can be reduced to a single one-dimensional integral expression [43]. Define

$\displaystyle c(u,\lambda)=\int_{-\infty}^u \; p_0(x) \; \exp(\lambda h(x)) \; dx$ (7.2)

$\displaystyle e(u,\lambda)=\int_u^{\infty} \; p_0(x) \; \exp(\lambda h(x)) \; dx$ (7.3)

We find the joint PDF of $ {\bf z}$ to be

$\displaystyle p_z({\bf z}) = \frac{N!}{D\pi} \left\{\prod_{m=1}^{M-1} \; p_0(z_...
... \exp\left[ -(\hat{\lambda}+iy) z_M \right] I(\hat{\lambda}+iy) \right\} \; dy,$ (7.4)

where

$\displaystyle D\stackrel{\mbox{\tiny$\Delta$}}{=}(t_1-1)! \; \left\{ \prod_{m=1}^{M-2}
\; (t_{m+1}-t_m -1)!\right\} (N-t_{M-1})!,
$

$\displaystyle I(\lambda) \stackrel{\mbox{\tiny$\Delta$}}{=}
e(z_1,\lambda)^{t_...
...ft[ e(z_{m+1},\lambda) -
e(z_{m},\lambda)\right]^{t_{m+1}-t_{m}-1}
\right\}.
$



Baggenstoss 2017-05-19