Imp.Estimates
Importance Sampling Estimates
Central moment, tail probability, and quantile estimates for a statistic under importance resampling.
- Keywords
- htest, nonparametric
Usage
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 toboot
ortilt.boot
. Use of these functions makes sense only when the bootstrap resampling used unequal weights for the observations. If the importance weightsw
are not supplied thenboot.out
is a required argument. It is also required ift
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 argumentt
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 att0[i]
. Ifimp.prob
is called without the argumentt0
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 inboot.out
). Eitherboot.out
ort
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 toimp.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 toimp.weights
to calculatew
.- 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 ifw
is supplied. Ifw
is missing thenq
is passed as an argument toimp.weights
and used to findw
.
Value
A list with the following components :
The alpha
levels used for the quantiles, if
imp.quantile
is used.
The values at which the tail probabilities are estimated, if
imp.prob
is used.
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
.
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
.
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
Examples
# NOT RUN {
# 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)
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)
# }