

   CCoonnffiiddeennccee EEnnvveellooppeess ffoorr CCuurrvveess

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

   AArrgguummeennttss::

   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
             then the first element gives the level for the
             pointwise envelope and the second gives the level
             for the overall envelope.

      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.

   DDeessccrriippttiioonn::

        This function calculates overall and pointwise confi-
        dence envelopes for a curve based on bootstrap repli-
        cates of the curve evaluated at a number of fixed
        points.

   DDeettaaiillss::

        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'.

   VVaalluuee::

        A list with the following components :

      point: A matrix with two rows corresponding to the values
             of the upper and lower pointwise confidence enve-
             lope at the same points as the bootstrap repli-
             cates were calculated.

    overall: A matrix similar to `point' but containing the
             envelope which controls the overall error.

       k.pt: The quantiles used for the pointwise envelope.

     err.pt: A 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.ov: The quantiles used for the overall envelope.

     err.ov: A vector with two components, the first gives the
             pointwise error rate for the overall envelope, and
             the second the overall error rate for that enve-
             lope.

    err.nom: A vector of length 2 giving the nominal error
             rates for the pointwise and the overall envelopes.

   RReeffeerreenncceess::

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

   SSeeee AAllssoo::

        `boot', `boot.ci'

   EExxaammpplleess::

        # 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)

