Normal Approximation Confidence Intervals
Using the normal approximation to a statistic, calculate equi-tailed two-sided confidence intervals.
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)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 |
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.
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.
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.
Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.
# 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.