\HeaderA{norm.ci}{Normal Approximation Confidence Intervals}{norm.ci}
\keyword{htest}{norm.ci}
\begin{Description}\relax
Using the normal approximation to a statistic, calculate equi-tailed two-sided 
confidence intervals.
\end{Description}
\begin{Usage}
\begin{verbatim}
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)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{boot.out}] A bootstrap output object returned from a call to \code{boot}.  If \code{t0} is
missing then \code{boot.out} is a required argument.  It is also required if
both \code{var.t0} and \code{t} are missing.

\item[\code{conf}] A scalar or vector containing the confidence level(s) of the required 
interval(s).

\item[\code{index}] The index of the statistic of interest within the output of a call to
\code{boot.out\$statistic}.  It is not used if \code{boot.out} is missing, in which
case \code{t0} must be supplied.

\item[\code{var.t0}] The variance of the statistic of interest.  If it is not supplied then 
\code{var(t)} is used.

\item[\code{t0}] The observed value of the statistic of interest.  If it is missing then it is
taken from \code{boot.out} which is required in that case.

\item[\code{t}] Bootstrap replicates of the variable of interest.  These are used to estimate 
the variance of the statistic of interest if \code{var.t0} is not supplied.  The
default value is \code{boot.out\$t[,index]}.

\item[\code{L}] The empirical influence values for the statistic of interest.  These are used
to calculate \code{var.t0} if neither \code{var.t0} nor \code{boot.out} are supplied.  If a
transformation is supplied through \code{h} then the influence values must be for the
untransformed statistic \code{t0}.

\item[\code{h}] A function defining a monotonic transformation,  the intervals are calculated
on the scale of \code{h(t)} and the inverse function \code{hinv} is applied to the 
resulting intervals.  \code{h} must be a function of one variable only and must
be vectorized. The default is the identity function.

\item[\code{hdot}] A function of one argument returning the derivative of \code{h}.  It is a required
argument if \code{h} is supplied and is used for approximating the variance of
\code{h(t0)}.  The default is the constant function 1.

\item[\code{hinv}] A function, like \code{h}, which returns the inverse of \code{h}.  It is used to transform
the intervals calculated on the scale of \code{h(t)} back to the original scale.
The default is the identity function.  If \code{h} is supplied but \code{hinv} is not,
then the intervals returned will be on the transformed scale.

\end{ldescription}
\end{Arguments}
\begin{Details}\relax
It is assumed that the statistic of interest has an approximately normal
distribution with variance \code{var.t0} and so a confidence interval of length
\code{2*qnorm((1+conf)/2)*sqrt(var.t0)} is found.  If \code{boot.out} or \code{t} are
supplied then the interval is bias-corrected using the bootstrap bias estimate,
and so the interval would be centred at \code{2*t0-mean(t)}.  Otherwise the interval
is centred at \code{t0}.
\end{Details}
\begin{Value}
If \code{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.
\end{Value}
\begin{Note}\relax
This function is primarily designed to be called by \code{boot.ci} to calculate 
the normal approximation after a bootstrap but it can also be used without
doing any bootstrap calculations as long as \code{t0} and \code{var.t0} can be supplied. See the examples below.
\end{Note}
\begin{References}\relax
Davison, A.C. and Hinkley, D.V. (1997) 
\emph{Bootstrap Methods and Their Application}. Cambridge University Press.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{boot.ci}{boot.ci}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
#  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)
\end{ExampleCode}
\end{Examples}

