Importance Sampling Estimates

Central moment, tail probability, and quantile estimates for a statistic under importance resampling.

imp.moments(boot.out = NULL, index = 1, t = boot.out$t[, index],
            w = NULL, def = TRUE, q = NULL)
imp.prob(boot.out = NULL, index = 1, t0 = boot.out$t0[index],
         t = boot.out$t[, index], w = NULL, def = TRUE, q = NULL)
imp.quantile(boot.out = NULL, alpha = NULL, index = 1,
             t = boot.out$t[, index], w = NULL, def = TRUE, q = NULL)

Arguments

boot.out	A object of class "boot" generated by a call to boot or tilt.boot. Use of these functions makes sense only when the bootstrap resampling used unequal weights for the observations. If the importance weights w are not supplied then boot.out is a required argument. It is also required if t is not supplied.
alpha	The alpha levels for the required quantiles. The default is to calculate the 1%, 2.5%, 5%, 10%, 90%, 95%, 97.5% and 99% quantiles.
index	The index of the variable of interest in the output of boot.out$statistic. This is not used if the argument t is supplied.
t0	The values at which tail probability estimates are required. For each value t0[i] the function will estimate the bootstrap cdf evaluated at t0[i]. If imp.prob is called without the argument t0 then the bootstrap cdf evaluated at the observed value of the statistic is found.
t	The bootstrap replicates of a statistic. By default these are taken from the bootstrap output object boot.out but they can be supplied separately if required (e.g. when the statistic of interest is a function of the calculated values in boot.out). Either boot.out or t must be supplied.
w	The importance resampling weights for the bootstrap replicates. If they are not supplied then boot.out must be supplied, in which case the importance weights are calculated by a call to imp.weights.
def	A logical value indicating whether a defensive mixture is to be used for weight calculation. This is used only if w is missing and it is passed unchanged to imp.weights to calculate w.
q	A vector of probabilities specifying the resampling distribution from which any estimates should be found. In general this would correspond to the usual bootstrap resampling distribution which gives equal weight to each of the original observations. The estimates depend on this distribution only through the importance weights w so this argument is ignored if w is supplied. If w is missing then q is passed as an argument to imp.weights and used to find w.

Value

A list with the following components :

alpha

The alpha levels used for the quantiles, if imp.quantile is used.

t0

The values at which the tail probabilities are estimated, if imp.prob is used.

raw

The raw importance resampling estimates. For imp.moments this has length 2, the first component being the estimate of the mean and the second being the variance estimate. For imp.prob, raw is of the same length as t0, and for imp.quantile it is of the same length as alpha.

rat

The ratio importance resampling estimates. In this method the weights w are rescaled to have average value one before they are used. The format of this vector is the same as raw.

reg

The regression importance resampling estimates. In this method the weights which are used are derived from a regression of t*w on w. This choice of weights can be shown to minimize the variance of the weights and also the Euclidean distance of the weights from the uniform weights. The format of this vector is the same as raw.

References

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

Hesterberg, T. (1995) Weighted average importance sampling and defensive mixture distributions. Technometrics, 37, 185--194.

Johns, M.V. (1988) Importance sampling for bootstrap confidence intervals. Journal of the American Statistical Association, 83, 709--714.

See also

boot, exp.tilt, imp.weights, smooth.f, tilt.boot

Examples

# Example 9.8 of Davison and Hinkley (1997) requires tilting the 
# resampling distribution of the studentized statistic to be centred 
# at the observed value of the test statistic, 1.84.  In this example
# we show how certain estimates can be found using resamples taken from
# the tilted distribution.
grav1 <- gravity[as.numeric(gravity[,2]) >= 7, ]
grav.fun <- function(dat, w, orig) {
     strata <- tapply(dat[, 2], as.numeric(dat[, 2]))
     d <- dat[, 1]
     ns <- tabulate(strata)
     w <- w/tapply(w, strata, sum)[strata]
     mns <- as.vector(tapply(d * w, strata, sum)) # drop names
     mn2 <- tapply(d * d * w, strata, sum)
     s2hat <- sum((mn2 - mns^2)/ns)
     c(mns[2] - mns[1], s2hat, (mns[2] - mns[1] - orig)/sqrt(s2hat))
}
grav.z0 <- grav.fun(grav1, rep(1, 26), 0)
grav.L <- empinf(data = grav1, statistic = grav.fun, stype = "w",
                 strata = grav1[,2], index = 3, orig = grav.z0[1])
grav.tilt <- exp.tilt(grav.L, grav.z0[3], strata = grav1[, 2])
grav.tilt.boot <- boot(grav1, grav.fun, R = 199, stype = "w",
                       strata = grav1[, 2], weights = grav.tilt$p,
                       orig = grav.z0[1])
# Since the weights are needed for all calculations, we shall calculate
# them once only.
grav.w <- imp.weights(grav.tilt.boot)
grav.mom <- imp.moments(grav.tilt.boot, w = grav.w, index = 3)
grav.p <- imp.prob(grav.tilt.boot, w = grav.w, index = 3, t0 = grav.z0[3])
unlist(grav.p)
#>        t0       raw       rat       reg 
#> 1.8401182 0.6654464 0.9622113 0.9759514 
grav.q <- imp.quantile(grav.tilt.boot, w = grav.w, index = 3,
                       alpha = c(0.9, 0.95, 0.975, 0.99))
as.data.frame(grav.q)
#>   alpha      raw      rat     reg
#> 1 0.900 2.458883 1.430694 3.47954
#> 2 0.950 2.871551 1.750000 3.47954
#> 3 0.975 3.102947 1.985627 3.47954
#> 4 0.990 3.183003 2.300204 3.47954