norm.ci {boot}  R Documentation 
Using the normal approximation to a statistic, calculate equitailed twosided 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 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.

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 biascorrected using the bootstrap bias estimate,
and so the interval would be centred at 2*t0mean(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 airconditioning 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)