# linear.approx

##### 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.

- Keywords
- nonparametric

##### Usage

```
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.

##### 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`

.

##### Value

A vector of length `boot.out$R`

with the linear approximations to the
statistic of interest for each of the bootstrap samples.

##### References

Davison, A.C. and Hinkley, D.V. (1997)
*Bootstrap Methods and Their Application*. Cambridge University Press.

##### See Also

##### Examples

```
# NOT RUN {
# 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)
# }
```

*Documentation reproduced from package boot, version 1.3-24, License: Unlimited*