# norm.ci: Normal Approximation Confidence Intervals In boot: Bootstrap Functions (Originally by Angelo Canty for S)

## Description

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

## Usage

 ```1 2 3``` ```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`.

## Value

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.

## 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.

## References

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

## See Also

`boot.ci`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```# 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) ```

### Example output

```     conf
[1,] 0.95 46.93055 169.2361
  61.38157 190.31782
conf
[1,] 0.95 1.167046 1.873579
```

boot documentation built on July 5, 2019, 5:02 p.m.