Concomitants variance reduction

DESCRIPTION:

Adjust a bootstrap object for greater accuracy.

USAGE:

concomitants.bootstrap(x, subset.statistic = <all>, 
    L = resampGetL(x), 
    group, treatment, 
    conv.factor = 0.1, 
    frame.eval = <<see below>>) 

REQUIRED ARGUMENTS:

x
a object.

OPTIONAL ARGUMENTS:

subset.statistic
vector of integers, indicating which columns of the bootstrap replicates to adjust.
L
matrix containing empirical influence values or other values to use in creating a linear approximation to the statistic being bootstrapped, with one column for each dimension (parameter) of the statistic.
conv.factor
scalar, convolution factor passed to qDiscreteMean. A nonzero value convolves the discrete distribution with a normal distribution; a small value like 0.1 improves stability, a larger value like n/(n-1) adds enough variance that the variance of the result is s^2/n where s is calculated with a divisor of (n-1).
group
vector defining stratified sampling. This is normally calculated automatically.
treatment
vector defining two groups. This is normally calculated automatically.
frame.eval
frame where the data and other objects used when creating x can be found. You need to specify this if objects can't be found by their original names, or have changed; see .

VALUE:

an object of class "concomitants" which inherits from "bootstrap". This is the same as x, except for components:
call
the call used to create the object.
replicates
the adjusted replicates.
L
the input L
estimate
adjusted estimate of mean, bias, and standard error.
original
list containing the components "call", "estimate" and "replicates" from x.

DETAILS:

This function adjusts the distribution of the bootstrap replicates, based on difference between the observed and exact distributions of a linear approximation. Given a vector L such as that created by (empirical influence function) or (influence, jackknife, or regression methods), this function will (1) evaluate the linear approximation for bootstrap samples, e.g. Lstar=indexMeans(L,resampGetIndices(x)) , (2) compute saddlepoint approximations for quantiles of the distribution of Lstar using , then (3) adjust the replicates component of the bootstrap object based on the observed values of the linear approximation for bootstrap samples and the corresponding quantiles.

This implementation uses smooth.spline to allow the relationship between Lstar and the statistic to be curvilinear. The higher the correlation between Lstar and a smooth monotone transformation of the statistic being bootstrapped the more accurate the result is.

REFERENCES:

Do, K. and Hall, P. (1992), "Distribution Estimation using Concomitants of Order Statistics, with Application to Monte Carlo Simulation for the Bootstrap," Journal of the Royal Statistical Society Series B, 54(2), 595-607.

Efron, B. (1990), "More Efficient Bootstrap Computations," Journal of the American Statistical Society, 85, 79-89.

Hesterberg, T.C. (1995), "Tail-Specific Linear Approximations for Efficient Bootstrap Simulations," Journal of Computational and Graphical Statistics, 4, 113-133.

Hesterberg, T.C. (1997), "Fast Bootstrapping by Combining Importance Sampling and Concomitants," Computing Science and Statistics, 29(2), 72-78.

SEE ALSO:

, , , .

For an annotated list of functions in the package, including other high-level resampling functions, see: .

EXAMPLES:

bfit <- bootstrap(qexp(ppoints(50)), mean) 
abfit <- concomitants(bfit) 
bfit # Bias should be zero, but differs due to random variation 
abfit # Bias is very near zero 
qqnorm(bfit) 
qqnorm(abfit) # much less random variation 
 
bfit <- bootstrap(1:50, var) 
abfit <- concomitants(bfit) 
bfit 
abfit 
plot(bfit) 
plot(abfit)