Control Variate Calculations

This function will find control variate estimates from a bootstrap output object. It can either find the adjusted bias estimate using post-simulation balancing or it can estimate the bias, variance, third cumulant and quantiles, using the linear approximation as a control variate.

control(boot.out, L = NULL, distn = NULL, index = 1, t0 = NULL,
        t = NULL, bias.adj = FALSE, alpha = NULL, ...)

Arguments

boot.out	A bootstrap output object returned from boot. The bootstrap replicates must have been generated using the usual nonparametric bootstrap.
L	The empirical influence values for the statistic of interest. If L is not supplied then empinf is called to calculate them from boot.out.
distn	If present this must be the output from smooth.spline giving the distribution function of the linear approximation. This is used only if bias.adj is FALSE. Normally this would be found using a saddlepoint approximation. If it is not supplied in that case then it is calculated by saddle.distn.
index	The index of the variable of interest in the output of boot.out$statistic.
t0	The observed value of the statistic of interest on the original data set boot.out$data. This argument is used only if bias.adj is FALSE. The input value is ignored if t is not also supplied. The default value is is boot.out$t0[index].
t	The bootstrap replicate values of the statistic of interest. This argument is used only if bias.adj is FALSE. The input is ignored if t0 is not supplied also. The default value is boot.out$t[,index].
bias.adj	A logical variable which if TRUE specifies that the adjusted bias estimate using post-simulation balance is all that is required. If bias.adj is FALSE (default) then the linear approximation to the statistic is calculated and used as a control variate in estimates of the bias, variance and third cumulant as well as quantiles.
alpha	The alpha levels for the required quantiles if bias.adj is FALSE.
...	Any additional arguments that boot.out$statistic requires. These are passed unchanged every time boot.out$statistic is called. boot.out$statistic is called once if bias.adj is TRUE, otherwise it may be called by empinf for empirical influence calculations if L is not supplied.

Value

If bias.adj is TRUE then the returned value is the adjusted bias estimate.

If bias.adj is FALSE then the returned value is a list with the following components

L

The empirical influence values used. These are the input values if supplied, and otherwise they are the values calculated by empinf.

tL

The linear approximations to the bootstrap replicates t of the statistic of interest.

bias

The control estimate of bias using the linear approximation to t as a control variate.

var

The control estimate of variance using the linear approximation to t as a control variate.

k3

The control estimate of the third cumulant using the linear approximation to t as a control variate.

quantiles

A matrix with two columns; the first column are the alpha levels used for the quantiles and the second column gives the corresponding control estimates of the quantiles using the linear approximation to t as a control variate.

distn

An output object from smooth.spline describing the saddlepoint approximation to the bootstrap distribution of the linear approximation to t. If distn was supplied on input then this is the same as the input otherwise it is calculated by a call to saddle.distn.

Details

If bias.adj is FALSE then the linear approximation to the statistic is found and evaluated at each bootstrap replicate. Then using the equation T* = Tl*+(T*-Tl*), moment estimates can be found. For quantile estimation the distribution of the linear approximation to t is approximated very accurately by saddlepoint methods, this is then combined with the bootstrap replicates to approximate the bootstrap distribution of t and hence to estimate the bootstrap quantiles of t.

References

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

Davison, A.C., Hinkley, D.V. and Schechtman, E. (1986) Efficient bootstrap simulation. Biometrika, 73, 555--566.

Efron, B. (1990) More efficient bootstrap computations. Journal of the American Statistical Association, 55, 79--89.

See also

boot, empinf, k3.linear, linear.approx, saddle.distn, smooth.spline, var.linear

Examples

# Use of control variates for the variance of the air-conditioning data
mean.fun <- function(d, i)
{    m <- mean(d$hours[i])
     n <- nrow(d)
     v <- (n-1)*var(d$hours[i])/n^2
     c(m, v)
}
air.boot <- boot(aircondit, mean.fun, R = 999)
control(air.boot, index = 2, bias.adj = TRUE)
#> [1] 5.207624
air.cont <- control(air.boot, index = 2)
# Now let us try the variance on the log scale.
air.cont1 <- control(air.boot, t0 = log(air.boot$t0[2]),
                     t = log(air.boot$t[, 2]))