pivot statistic

We define , and the corresponding z-value We denote this z-value as z8[v]=zc[v,{0,0,0,0},0,1].

We slightly alter z8[v] and denoted zq[v]= below.

Here we define a function to collect 's for later use

check if this is correct.

Now we continue to calculate the distribution function of zq[v]

The distribution function of zq is obtained here. =Φ{zfzq[w,{q0,q1,q2,q3},v0,tau]}.

For tau=1, zfzq becomes

When v0=0, tau=1,  zfzq becomes

In particular, the distribution function of z8 under v0=0 is Pr{Z8<=w;0,1)=Φ(w).