Bootstrap Methods

In this part, the asymptotic accuracies of several bootstrap methods are discussed. We use the zc-formula obtained in the previous part.

Startup

This section initializes the Mathematica session.

Asymptotic Analysis of Bootstrap Methods

This section calculates the distribution functions of several bootstrap methods for showing their asymptotic accuracies in terms of the unbiasedness of hypothesis testing of the region R. The calculations are based on the zc-formula obtained in the previous part. We first define a pivot statistic and shows its third-order accuracy. The bootstrap probability is first-order accurate, the double bootstrap is second-order accurate, and the two-level bootstrap is second-order accurate.

preliminary

The zc-formula is given in zc[w,cc,v0,tau]=zformtau=, where cc={c0,c1,c2,c3} specifies the modified signed distance w. The signed distance v is expressed in terms of w by  , or in the inverse series . We write the coefficients or , etc. The rule to calculate {c0,c1,c2,c3} in terms of {cb0,cb1,cb2,cb3} is given in "rulecc2cb" or in cb2cc function. The function zq2cc[exp] calculates the {c0,c1,c2,c3} from the polynomial of w in terms of v.

the pivot and some existing bootstrap methods

The pivot is defined as z8[v]==zc[v,{0,0,0,0},0,1]. 's and 's are in cbz8 and ccz8 respectively for z8[v].  We define zq[v]==z8[v] + q0 + q1 v + q2 + q3 , where the coefficients qq={q0,q1,q2,q3} specify the zq[v]. zq2qq calculates qq from any z-value. 's and 's are in cbzq and cczq respectively for zq[v]. The distribution function of zq is obtained as =Φ{zfzq[w,{q0,q1,q2,q3},v0,tau]}. We observe that zfzq[w,{0,0,0,0},0,1]=w, and thus the distribution function of z8 under v0=0 is Pr{Z8<=w;0,1)=Φ(w).

The bootstrap probability is for y=η(0,v), and the corresponding z-value is z1[v,tau1]==-zc[0,{0,0,0,0},v,tau1]. For tau1=1, we define z0[v]=z1[v,1], which can be regarded as another w. For general tau1, w1=tau1 z1[v,tau1] is regarded as another w with 's being cbw1 and 's being ccw1, and the distribution function is expressed as . For tau=1 and tau1=1, we have , which becomes under v0=0, showing the first-order accuracy of z0[v].

The z-value of the double bootstrap probability is , and we observe that z8[v]=zd[v], showing the double bootstrap asymptotically equivalent to the third-order accurate pivot statistic up to terms.

The ABC formula is given in abcformula[v,ac]. The z-value of the two-level bootstrap method is calculated in za[v]. Its 's are in qqza. The distribution function of za[v]  under tau=1 is Pr{za[V]<=w;v0,1}=Φ[zfzq[w,qqza,v0,1]]. This becomes for v0=0, showing the second-order accuracy of the two-level bootstrap.

multistep-multiscale bootstrap method

Here we calculate the asymptotic accuracy of the three-step multiscale bootstrap method.

The pivot z8[v] is generalized to define  , which reduces to z8[v,0,1]=z8[v]. We standardize z8[v,v0,tau] by w8[v,v0,tau]=z8[v,v0,tau]*isz8 + v0, where , so that .  The distribution function is We show .
We obtain the inverse function of z8[v,v0,tau]=z in terms of v so that v8[z,v0,tau]=v is defined.

Let func[z]=(az+b) + rem[z], where rem[z]=is of order We would like to calculate intfz=, where f, F, and Q are, respectively, the density, the distribution, and quantile functions of the standard normal distribution. Let us define . Then we will show that  intfz=ddint[a,b,d,z]=, where d=func2dd[a,b,rem,z] calculates the coefficients of the expansion .

Using ddint function defined above, we will calculate   =. This is the z-value of the twostep-multiscale bootstrap probability.

Similarly we will calculate the z-value of the threestep-multiscale bootstrap probability defined by
  =.

The six geometric quantities are denoted by G1,...,G6 here. The scaling parameter are denoted by S1,...,S4 here. We define Z3G and Z8G in terms of G1,...,G6, S1,...,S4, and will show that Z3G=z3[v,tau1,tau2,tau3] and Z8G=z8[v].