boott function

Bootstrap-t Confidence Limits

See Efron and Tibshirani (1993) for details on this function.

boott(x,theta, ..., sdfun=sdfunboot, nbootsd=25, nboott=200,
        VS=FALSE, v.nbootg=100, v.nbootsd=25, v.nboott=200,
        perc=c(.001,.01,.025,.05,.10,.50,.90,.95,.975,.99,.999))

Arguments

• x: a vector containing the data. Nonparametric bootstrap sampling is used. To bootstrap from more complex data structures (e.g. bivariate data) see the last example below.

• theta: function to be bootstrapped. Takes x as an argument, and may take additional arguments (see below and last example).

• ...: any additional arguments to be passed to theta

• sdfun: optional name of function for computing standard deviation of theta based on data x. Should be of the form: sdmean <- function(x,nbootsd,theta,...) where nbootsd

  is a dummy argument that is not used. If theta is the mean, for example, sdmean <- function(x,nbootsd,theta,...)

  {sqrt(var(x)/length(x))} . If sdfun is missing, then boott uses an inner bootstrap loop to estimate the standard deviation of theta(x)

• nbootsd: The number of bootstrap samples used to estimate the standard deviation of theta(x)

• nboott: The number of bootstrap samples used to estimate the distribution of the bootstrap T statistic. 200 is a bare minimum and 1000 or more is needed for reliable $\alpha \%$ confidence points, $\alpha > .95$ say. Total number of bootstrap samples is nboott*nbootsd.

• VS: If TRUE, a variance stabilizing transformation is estimated, and the interval is constructed on the transformed scale, and then is mapped back to the original theta scale. This can improve both the statistical properties of the intervals and speed up the computation. See the reference Tibshirani (1988) given below. If FALSE, variance stabilization is not performed.

• v.nbootg: The number of bootstrap samples used to estimate the variance stabilizing transformation g. Only used if VS=TRUE.

• v.nbootsd: The number of bootstrap samples used to estimate the standard deviation of theta(x). Only used if VS=TRUE.

• v.nboott: The number of bootstrap samples used to estimate the distribution of the bootstrap T statistic. Only used if VS=TRUE. Total number of bootstrap samples is v.nbootg*v.nbootsd + v.nboott.

• perc: Confidence points desired.

Returns

list with the following components: - confpoints: Estimated confidence points

• theta, g: theta

  and g are only returned if VS=TRUE was specified. (theta[i],g[i]), i=1,length(theta)

  represents the estimate of the variance stabilizing transformation g at the points theta[i].

• call: The deparsed call

References

Tibshirani, R. (1988) "Variance stabilization and the bootstrap". Biometrika (1988) vol 75 no 3 pages 433-44.

Hall, P. (1988) Theoretical comparison of bootstrap confidence intervals. Ann. Statisi. 16, 1-50.

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

#  estimated confidence points for the mean
   x <- rchisq(20,1)
   theta <- function(x){mean(x)}
   results <- boott(x,theta)
# estimated confidence points for the mean, 
#  using variance-stabilization bootstrap-T method
   results <-  boott(x,theta,VS=TRUE)
   results$confpoints          # gives confidence points
# plot the estimated var stabilizing transformation
   plot(results$theta,results$g) 
# use standard formula for stand dev of mean
# rather than an inner bootstrap loop
   sdmean <- function(x, ...) 
       {sqrt(var(x)/length(x))}
   results <-  boott(x,theta,sdfun=sdmean) 
                                     
# To bootstrap functions of more  complex data structures, 
# write theta so that its argument x
#  is the set of observation numbers  
#  and simply  pass as data to boot the vector 1,2,..n. 
# For example, to bootstrap
# the correlation coefficient from a set of 15 data pairs:                              
    xdata <- matrix(rnorm(30),ncol=2)
    n <- 15
    theta <- function(x, xdata){ cor(xdata[x,1],xdata[x,2]) }
    results <- boott(1:n,theta, xdata)