# Smooth Distributions on Data Points 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`. ```r 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$R` and the values must be in the same order as the bootstrap replicates in `boot.out`. - `width`: The standardized width for the kernel smoothing. The smoothing uses a value of `width*s` for epsilon, where `s` is the bootstrap estimate of the standard error of the statistic of interest. `width` should take a value in the range (0.2, 1) to produce a reasonable smoothed distribution. If `width` is too large then the distribution becomes closer to uniform. ## Returns If `length(theta)` is 1 then a vector with the same length as the data set `boot.out$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`. ## 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). ## 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. ## See Also `boot`, `exp.tilt`, `tilt.boot` ## Examples ```r # 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). ```

smooth.f function