# Linear Approximation of Bootstrap Replicates This function takes a bootstrap object and for each bootstrap replicate it calculates the linear approximation to the statistic of interest for that bootstrap sample. ```r linear.approx(boot.out, L = NULL, index = 1, type = NULL, t0 = NULL, t = NULL, ...) ``` ## Arguments - `boot.out`: An object of class `"boot"` representing a nonparametric bootstrap. It will usually be created by the function `boot`. - `L`: A vector containing the empirical influence values for the statistic of interest. If it is not supplied then `L` is calculated through a call to `empinf`. - `index`: The index of the variable of interest within the output of `boot.out$statistic`. - `type`: This gives the type of empirical influence values to be calculated. It is not used if `L` is supplied. The possible types of empirical influence values are described in the help for `empinf`. - `t0`: The observed value of the statistic of interest. The input value is used only if one of `t` or `L` is also supplied. The default value is `boot.out$t0[index]`. If `t0` is supplied but neither `t` nor `L` are supplied then `t0` is set to `boot.out$t0[index]` and a warning is generated. - `t`: A vector of bootstrap replicates of the statistic of interest. If `t0` is missing then `t` is not used, otherwise it is used to calculate the empirical influence values (if they are not supplied in `L`). - `...`: Any extra arguments required by `boot.out$statistic`. These are needed if `L` is not supplied as they are used by `empinf` to calculate empirical influence values. ## Returns A vector of length `boot.out$R` with the linear approximations to the statistic of interest for each of the bootstrap samples. ## Details The linear approximation to a bootstrap replicate with frequency vector `f` is given by `t0 + sum(L * f)/n` in the one sample with an easy extension to the stratified case. The frequencies are found by calling `boot.array`. ## References Davison, A.C. and Hinkley, D.V. (1997) **Bootstrap Methods and Their Application**. Cambridge University Press. ## See Also `boot`, `empinf`, `control` ## Examples ```r # Using the city data let us look at the linear approximation to the # ratio statistic and its logarithm. We compare these with the # corresponding plots for the bigcity data ratio <- function(d, w) sum(d$x * w)/sum(d$u * w) city.boot <- boot(city, ratio, R = 499, stype = "w") bigcity.boot <- boot(bigcity, ratio, R = 499, stype = "w") op <- par(pty = "s", mfrow = c(2, 2)) # The first plot is for the city data ratio statistic. city.lin1 <- linear.approx(city.boot) lim <- range(c(city.boot$t,city.lin1)) plot(city.boot$t, city.lin1, xlim = lim, ylim = lim, main = "Ratio; n=10", xlab = "t*", ylab = "tL*") abline(0, 1) # Now for the log of the ratio statistic for the city data. city.lin2 <- linear.approx(city.boot,t0 = log(city.boot$t0), t = log(city.boot$t)) lim <- range(c(log(city.boot$t),city.lin2)) plot(log(city.boot$t), city.lin2, xlim = lim, ylim = lim, main = "Log(Ratio); n=10", xlab = "t*", ylab = "tL*") abline(0, 1) # The ratio statistic for the bigcity data. bigcity.lin1 <- linear.approx(bigcity.boot) lim <- range(c(bigcity.boot$t,bigcity.lin1)) plot(bigcity.lin1, bigcity.boot$t, xlim = lim, ylim = lim, main = "Ratio; n=49", xlab = "t*", ylab = "tL*") abline(0, 1) # Finally the log of the ratio statistic for the bigcity data. bigcity.lin2 <- linear.approx(bigcity.boot,t0 = log(bigcity.boot$t0), t = log(bigcity.boot$t)) lim <- range(c(log(bigcity.boot$t),bigcity.lin2)) plot(bigcity.lin2, log(bigcity.boot$t), xlim = lim, ylim = lim, main = "Log(Ratio); n=49", xlab = "t*", ylab = "tL*") abline(0, 1) par(op) ```

linear.approx function