R: Control Variate Calculations

control {boot} | R Documentation

## Control Variate Calculations

### Description

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.

### Usage

    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.

### 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`.

### 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`.

### 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)
    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]))

[Package boot version 1.2-30 Index]