a generalization of the pivot

We define a generalization of the pivot by . We use z8 in the analysis of the multistep bootstrap.

This reduces to the pivot when v0=0, tau=1.

We standardize z8[v,v0,tau] so that it can be regarded as w with proper coefficients. First, we find the rescaling factor.

The standardization of z8[v,v0,tau] up to term is denoted w8[v,v0,tau]=z8[v,v0,tau]*isz8 + v0.

Then the distribution function of W8=w8[V,v0,tau] is

Then, . The following equation implies that  .

We obtain the inverse function of z8[v,v0,tau]=z in terms of v so that v8[z,v0,tau]=v by applying the inversion formula of the modified signed distance to w8. We use v8-function in the following section.

Check if z8[v8[z]]=z and v8[z8[v]]=v