R: Examples for 'boot::smooth.f'

### ** 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 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] 0.01803607

imp.prob(grav.boot2, t0 = -grav.z0[3], t = -grav.boot2$t[, 3])

$t0
[1] -1.840118

$raw
[1] 0.01974692

$rat
[1] 0.01925778

$reg
[1] 0.01983488

# 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] -0.005652243

# can give very strange results (negative probabilities).

Examples for 'boot::smooth.f'

Smooth Distributions on Data Points