smooth.f {boot} R Documentation

## Smooth Distributions on Data Points

### Description

This 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.

### References

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.

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

# 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
# can give very strange results (negative probabilities).


[Package boot version 1.3-30 Index]