boot (version 1.1-6)

envelope: Confidence Envelopes for Curves

Description

This function calculates overall and pointwise confidence envelopes for a curve based on bootstrap replicates of the curve evaluated at a number of fixed points.

Usage

envelope(boot.out=NULL, mat=boot.out$t, level=0.95, index=1:ncol(mat))

Arguments

boot.out
An object of class "boot" for which boot.out$t contains the replicates of the curve at a number of fixed points.
mat
A matrix of bootstrap replicates of the values of the curve at a number of fixed points. This is a required argument if boot.out is not supplied and is set to boot.out$t otherwise.
level
The confidence level of the envelopes required. The default is to find 95% confidence envelopes. It can be a scalar or a vector of length 2. If it is scalar then both the pointwise and the overall envelopes are found at that level. If is a vector the
index
The numbers of the columns of mat which contain the bootstrap replicates. This can be used to ensure that other statistics which may have been calculated in the bootstrap are not considered as values of the function.

Value

  • A list with the following components :
  • pointA matrix with two rows corresponding to the values of the upper and lower pointwise confidence envelope at the same points as the bootstrap replicates were calculated.
  • overallA matrix similar to point but containing the envelope which controls the overall error.
  • k.ptThe quantiles used for the pointwise envelope.
  • err.ptA vector with two components, the first gives the pointwise error rate for the pointwise envelope, and the second the overall error rate for that envelope.
  • k.ovThe quantiles used for the overall envelope.
  • err.ovA vector with two components, the first gives the pointwise error rate for the overall envelope, and the second the overall error rate for that envelope.
  • err.nomA vector of length 2 giving the nominal error rates for the pointwise and the overall envelopes.

Details

The pointwise envelope is found by simply looking at the quantiles of the replicates at each point. The overall error for that envelope is then calculated using equation (4.17) of Davison and Hinkley (1997). A sequence of pointwise envelopes is then found until one of them has overall error approximately equal to the level required. If no such envelope can be found then the envelope returned will just contain the extreme values of each column of mat.

References

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

See Also

boot, boot.ci

Examples

Run this code
# Testing whether the final series of measurements of the gravity data
# may come from a normal distribution.  This is done in Examples 4.7 
# and 4.8 of Davison and Hinkley (1997).
data(gravity)
grav1 <- gravity$g[gravity$series==8]
grav.z <- (grav1-mean(grav1))/sqrt(var(grav1))
grav.gen <- function(dat,mle)
     rnorm(length(dat))
grav.qqboot <- boot(grav.z,sort,R=999,sim="parametric",ran.gen=grav.gen)
grav.qq <- qqnorm(grav.z,plot=F)
grav.qq <- lapply(grav.qq,sort)
plot(grav.qq,ylim=c(-3.5,3.5),ylab="Studentized Order Statistics",
     xlab="Normal Quantiles")
grav.env <- envelope(grav.qqboot,level=0.9)
lines(grav.qq$x,grav.env$point[1,],lty=4)
lines(grav.qq$x,grav.env$point[2,],lty=4)
lines(grav.qq$x,grav.env$overall[1,],lty=1)
lines(grav.qq$x,grav.env$overall[2,],lty=1)

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