smooth.f.RdThis function uses the method of frequency smoothing to find a distribution
on a data set which has a required value, theta, of the statistic of
interest. The method results in distributions which vary smoothly with
theta.
smooth.f(theta, boot.out, index = 1, t = boot.out$t[, index],
width = 0.5)The required value for the statistic of interest. If theta is a vector,
a separate distribution will be found for each element of theta.
A bootstrap output object returned by a call to boot.
The index of the variable of interest in the output of boot.out$statistic.
This argument is ignored if t is supplied. index must be a scalar.
The bootstrap values of the statistic of interest. This must be a vector of
length boot.out$R and the values must be in the same order as the bootstrap
replicates in boot.out.
The standardized width for the kernel smoothing. The smoothing uses a
value of width*s for epsilon, where s is the bootstrap estimate of the
standard error of the statistic of interest. width should take a value in
the range (0.2, 1) to produce a reasonable
smoothed distribution. If width is too large then the distribution becomes
closer to uniform.
If length(theta) is 1 then a vector with the same length as the data set
boot.out$data is returned. The value in position i is the probability
to be given to the data point in position i so that the distribution has
parameter value approximately equal to theta.
If length(theta) is bigger than 1 then the returned value is a matrix with
length(theta) rows each of which corresponds to a distribution with the
parameter value approximately equal to the corresponding value of theta.
The new distributional weights are found by applying a normal kernel smoother
to the observed values of t weighted by the observed frequencies in the
bootstrap simulation. The resulting distribution may not have
parameter value exactly equal to the required value theta but it will
typically have a value which is close to theta. The details of how this
method works can be found in Davison, Hinkley and Worton (1995) and Section
3.9.2 of Davison and Hinkley (1997).
Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.
Davison, A.C., Hinkley, D.V. and Worton, B.J. (1995) Accurate and efficient construction of bootstrap likelihoods. Statistics and Computing, 5, 257–264.
# 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 the book exponential tilting was used
# but it is also possible to use smooth.f.
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.boot <- boot(grav1, grav.fun, R = 499, stype = "w",
strata = grav1[, 2], orig = grav.z0[1])
grav.sm <- smooth.f(grav.z0[3], grav.boot, index = 3)
# Now we can run another bootstrap using these weights
grav.boot2 <- boot(grav1, grav.fun, R = 499, stype = "w",
strata = grav1[, 2], orig = grav.z0[1],
weights = grav.sm)
# Estimated p-values can be found from these as follows
mean(grav.boot$t[, 3] >= grav.z0[3])
#> [1] 0.02004008
imp.prob(grav.boot2, t0 = -grav.z0[3], t = -grav.boot2$t[, 3])
#> $t0
#> [1] -1.840118
#>
#> $raw
#> [1] 0.02216074
#>
#> $rat
#> [1] 0.02245096
#>
#> $reg
#> [1] 0.02208658
#>
# Note that for the importance sampling probability we must
# multiply everything by -1 to ensure that we find the correct
# probability. Raw resampling is not reliable for probabilities
# greater than 0.5. Thus
1 - imp.prob(grav.boot2, index = 3, t0 = grav.z0[3])$raw
#> [1] 0.0350879
# can give very strange results (negative probabilities).