# norm.ci {boot}

### Description

Using the normal approximation to a statistic, calculate equi-tailed two-sided confidence intervals.

### Usage

norm.ci(boot.out = NULL, conf = 0.95, index = 1, var.t0 = NULL, t0 = NULL, t = NULL, L = NULL, h = function(t) t, hdot = function(t) 1, hinv = function(t) t)

### Arguments

- boot.out
- A bootstrap output object returned from a call to
`boot`

. If`t0`

is missing then`boot.out`

is a required argument. It is also required if both`var.t0`

and`t`

are missing. - conf
- A scalar or vector containing the confidence level(s) of the required interval(s).
- index
- The index of the statistic of interest within the output of a call to
`boot.out$statistic`

. It is not used if`boot.out`

is missing, in which case`t0`

must be supplied. - var.t0
- The variance of the statistic of interest. If it is not supplied then
`var(t)`

is used. - t0
- The observed value of the statistic of interest. If it is missing then it is taken from
`boot.out`

which is required in that case. - t
- Bootstrap replicates of the variable of interest. These are used to estimate the variance of the statistic of interest if
`var.t0`

is not supplied. The default value is`boot.out$t[,index]`

. - L
- The empirical influence values for the statistic of interest. These are used to calculate
`var.t0`

if neither`var.t0`

nor`boot.out`

are supplied. If a transformation is supplied through`h`

then the influence values must be for the untransformed statistic`t0`

. - h
- A function defining a monotonic transformation, the intervals are calculated on the scale of
`h(t)`

and the inverse function`hinv`

is applied to the resulting intervals.`h`

must be a function of one variable only and must be vectorized. The default is the identity function. - hdot
- A function of one argument returning the derivative of
`h`

. It is a required argument if`h`

is supplied and is used for approximating the variance of`h(t0)`

. The default is the constant function 1. - hinv
- A function, like
`h`

, which returns the inverse of`h`

. It is used to transform the intervals calculated on the scale of`h(t)`

back to the original scale. The default is the identity function. If`h`

is supplied but`hinv`

is not, then the intervals returned will be on the transformed scale.

### Details

It is assumed that the statistic of interest has an approximately normal distribution with variance `var.t0`

and so a confidence interval of length `2*qnorm((1+conf)/2)*sqrt(var.t0)`

is found. If `boot.out`

or `t`

are supplied then the interval is bias-corrected using the bootstrap bias estimate, and so the interval would be centred at `2*t0-mean(t)`

. Otherwise the interval is centred at `t0`

.

### Values

If `length(conf)`

is 1 then a vector containing the confidence level and the endpoints of the interval is returned. Otherwise, the returned value is a matrix where each row corresponds to a different confidence level.

### References

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

### Note

This function is primarily designed to be called by `boot.ci`

to calculate the normal approximation after a bootstrap but it can also be used without doing any bootstrap calculations as long as `t0`

and `var.t0`

can be supplied. See the examples below.

### See Also

### Examples

# In Example 5.1 of Davison and Hinkley (1997), normal approximation # confidence intervals are found for the air-conditioning data. air.mean <- mean(aircondit$hours) air.n <- nrow(aircondit) air.v <- air.mean^2/air.n norm.ci(t0 = air.mean, var.t0 = air.v) exp(norm.ci(t0 = log(air.mean), var.t0 = 1/air.n)[2:3]) # Now a more complicated example - the ratio estimate for the city data. ratio <- function(d, w) sum(d$x * w)/sum(d$u *w) city.v <- var.linear(empinf(data = city, statistic = ratio)) norm.ci(t0 = ratio(city,rep(0.1,10)), var.t0 = city.v)

Documentation reproduced from package boot, version 1.3-9. License: Unlimited