Smooth Distributions on Data Points

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.

Usage

smooth.f(theta, boot.out, index = 1, t = boot.out$t[, index],
         width = 0.5)

Arguments

`theta`	The required value for the statistic of interest. If `theta` is a vector, a separate distribution will be found for each element of `theta`.
`boot.out`	A bootstrap output object returned by a call to `boot`.
`index`	The index of the variable of interest in the output of `boot.out$statistic`. This argument is ignored if `t` is supplied. `index` must be a scalar.
`t`	The bootstrap values of the statistic of interest. This must be a vector of length `boot.out$R` and the values must be in the same order as the bootstrap replicates in `boot.out`.
`width`	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.

Details

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).

Value

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.

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.

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 1.3-22 Index]