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In this vignette, we extend the analysis in Example 2 of Table 11.4, p. 423, of Boos and Stefanski (2013), “Essential Statistical Inference.” Table 11.4 compares three methods of estimating the variance of the 20% trimmed mean. The three methods are the Influence Curve Method (Delta Method), the jackknife, and the bootstrap. The main entries of the table are based on N Monte Carlo samples of size n=20, leading to a matrix of three sets, one set for each distribution: the N 20% trimmed means and 3 variance estimators. Here we focus on the 3 columns of Table 11.4 based on
CV = the coefficient of variation of the N variance estimators
Under the uniform distribution, the estimated CVs are 0.41, 0.34, and 0.38. With an approximate SE of 0.01, these three CVs are reasonably well-separated (“significantly different”). However, for the other two distributions, the CVs are not as well-separeated compared to their SEs. In such a case, one might be interested in the standard errors of the differences or of the ratios of the CVs, because the these latter standard errors should be less than if the summaries were independent. The function pairwise.se is designed to get all pairwise SEs.
Here we display the first line of the function:
pairwise.se <- function(x,xcol,diff=TRUE,digits=4,B=0,seed=NULL,summary.f,...){We see the main arguments are the Monte Carlo raw output data
x, the columns xcol of x to use,
and the summary summary.f. Here, summary.f is
just a simple function of one vector as used in
mc.se.vector. In the example below, we use
summary.f=CV.
In the Simulation Code below, we generate
an N=1,000 by 12 matrix hold of Monte Carlo outout. Columns
1-4 of hold are for the uniform(0,1) distribution, columns
5-8 are for the standard normal distribution, and columns 9-12 are for
the standard exponential distribution. Columns 1, 4, and 8 are 20%
trimmed means, and the other columns are estimated variances.
To illustrate pairwise.se, we
first get the standard errors of the differences of CV values in Table
11.4 for the normal distribution using columns 6-8 of the output data
frame hold.
> pairwise.se(hold,xcol=6:8,summary.f=cv)
elem summi summj summary se t.stat N method
1 6 7 0.4695 0.4232 0.0463 0.0072 6.4254 1000 Jackknife
2 6 8 0.4695 0.4359 0.0337 0.0090 3.7556 1000 Jackknife
3 7 8 0.4232 0.4359 -0.0126 0.0071 -1.7795 1000 JackknifeFirst notice that the summaries (0.47, 0.42, 0.43) for the normal
distribution in Table 11.4
show up under summi and summj. Then their
differences are listed under summary. The standard errors
of the differences are under se. Next, t.stat
is summary/se if differences are used and
(summary-1)/se if ratios are used (diff=F). To
illustrate the use of ratios,
> pairwise.se(hold,xcol=6:8,diff=F,summary.f=cv)
elem summi summj summary se t.stat N method
1 6 7 0.4695 0.4232 1.1095 0.0178 6.1676 1000 Jackknife
2 6 8 0.4695 0.4359 1.0773 0.0216 3.5719 1000 Jackknife
3 7 8 0.4232 0.4359 0.9710 0.0160 -1.8174 1000 JackknifeNotice that summi and summj are the same as
for differences, but now summary holds the ratios. The t
statistics are similar to those for differences.
If the bootstrap is to be used in place of the jackknife, we have
> pairwise.se(hold,xcol=6:8,B=1000,seed=769,summary.f=cv)
elem summi summj summary se t.stat B seed N method
1 6 7 0.4695 0.4232 0.0463 0.0070 6.5859 1000 769 1000 Bootstrap
2 6 8 0.4695 0.4359 0.0337 0.0090 3.7520 1000 769 1000 Bootstrap
3 7 8 0.4232 0.4359 -0.0126 0.0068 -1.8640 1000 769 1000 Bootstrap
> pairwise.se(hold,xcol=6:8,diff=F,B=1000,seed=769,summary.f=cv)
elem summi summj summary se t.stat B seed N method
1 6 7 0.4695 0.4232 1.1095 0.0178 6.1481 1000 769 1000 Bootstrap
2 6 8 0.4695 0.4359 1.0773 0.0211 3.6563 1000 769 1000 Bootstrap
3 7 8 0.4232 0.4359 0.9710 0.0160 -1.8179 1000 769 1000 BootstrapThe following code creates an N=1,000 by 12 matrix ‘hold’ of Monte Carlo outout. Columns 1-4 are for the uniform(0,1) distribution, columns 5-8 are for the standard normal distribution, and columns 9-12 are for the standard exponential distribution. Columns 1, 4, and 8 are 20% trimmed means, and the other columns are estmated variances.
trim20 <- function(x){mean(x,.2)} # 20% trimmed mean function
trim.var <- function(x,trim){ # var. est. for trim20
n <- length(x);h<-floor(trim*n)
tm <- mean(x,trim);sort(x)->x
ss <- h*((x[h+1]-tm)^2+(x[n-h]-tm)^2)
ss <- ss+sum((x[(h+1):(n-h)]-tm)^2)
ss/((n-2*h)*(n-2*h-1))
}
trim20.var<-function(x){trim.var(x,.2)}
N=1000
hold <- matrix(0,nrow=N,ncol=12)
set.seed(351)
z <- matrix(runif(N*20),nrow=N)
hold[,1] <- apply(z,1,trim20)
hold[,2] <- apply(z,1,trim20.var)
hold[,3] <- apply(z,1,jack.var,theta=trim20)
hold[,4] <- apply(z,1,boot.var,theta=trim20,B=100)
set.seed(352)
z<- matrix(rnorm(N*20),nrow=N)
hold[,5] <- apply(z,1,trim20)
hold[,6] <- apply(z,1,trim20.var)
hold[,7] <- apply(z,1,jack.var,theta=trim20)
hold[,8] <- apply(z,1,boot.var,theta=trim20,B=100)
set.seed(353)
z<- matrix(rexp(N*20),nrow=N)
hold[,9] <- apply(z,1,trim20)
hold[,10] <- apply(z,1,trim20.var)
hold[,11] <- apply(z,1,jack.var,theta=trim20)
hold[,12] <- apply(z,1,boot.var,theta=trim20,B=100)These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.