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 [51]. Define

$$c(u,\lambda)=\int_{-\infty}^u \; p_0(x) \; \exp(\lambda h(x)) \; dx$$ (8.2)

$$e(u,\lambda)=\int_u^{\infty} \; p_0(x) \; \exp(\lambda h(x)) \; dx$$ (8.3)

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

$$p_z({\bf z}) = \frac{N!}{D\pi} \left\{\prod_{m=1}^{M-1} \; p_0(z_... ... \exp\left[ -(\hat{\lambda}+iy) z_M \right] I(\hat{\lambda}+iy) \right\} \; dy,$$ (8.4)

where

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

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