norm.ci
Normal Approximation Confidence Intervals
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 |
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 |
var.t0 | The variance of the statistic of interest. If it is not supplied then |
t0 | The observed value of the statistic of interest. If it is missing then it is taken from |
t | Bootstrap replicates of the variable of interest. These are used to estimate the variance of the statistic of interest if |
L | The empirical influence values for the statistic of interest. These are used to calculate |
h | A function defining a monotonic transformation, the intervals are calculated on the scale of |
hdot | A function of one argument returning the derivative of |
hinv | A function, like |
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
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)
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Licensed under the GNU General Public License.