\HeaderA{tilt.boot}{Non-parametric Tilted Bootstrap}{tilt.boot}
\keyword{nonparametric}{tilt.boot}
\begin{Description}\relax
This function will run an initial bootstrap with equal resampling 
probabilities (if required) and will use the output of the initial run to 
find resampling probabilities which put the
value of the statistic at required values.  It then runs an importance
resampling bootstrap using the calculated probabilities as the resampling
distribution.
\end{Description}
\begin{Usage}
\begin{verbatim}
tilt.boot(data, statistic, R, sim="ordinary", stype="i", 
          strata=rep(1, n), L=NULL, theta=NULL, 
          alpha=c(0.025, 0.975), tilt=TRUE, width=0.5, 
          index=1, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{data}] The data as a vector, matrix or data frame.  If it is a matrix or data frame 
then each row is considered as one (multivariate) observation.

\item[\code{statistic}] A function which when applied to data returns a vector containing the
statistic(s) of interest.  It must take at least two arguments.  The first
argument will always be \code{data} and the second should be a vector of indices,
weights or frequencies describing the bootstrap sample.  Any other arguments
must be supplied to \code{tilt.boot} and will be passed unchanged to statistic each
time it is called.

\item[\code{R}] The number of bootstrap replicates required.  This will generally be a vector,
the first value stating how many uniform bootstrap simulations are to be 
performed at the initial stage.  The remaining values of \code{R} are the number of 
simulations to be performed resampling from each reweighted distribution.  
The first value of
\code{R} must always be present, a value of 0 implying that no uniform resampling is
to be carried out.  Thus \code{length(R)} should always equal \code{1+length(theta)}.

\item[\code{sim}] This is a character string indicating the type of bootstrap simulation required.
There are only two possible values that this can take: \code{"ordinary"} and
\code{"balanced"}.  If other simulation types are required for the initial
un-weighted bootstrap then it will be necessary to run
\code{boot}, calculate the weights appropriately, and run \code{boot} again using the
calculated weights.

\item[\code{stype}] A character string indicating the type of second argument expected by 
\code{statistic}.  The possible values that \code{stype} can take are \code{"i"} (indices), 
\code{"w"} (weights) and \code{"f"} (frequencies).

\item[\code{strata}] An integer vector or factor representing the strata for multi-sample problems.

\item[\code{L}] The empirical influence values for the statistic of interest.  They are used
only for exponential tilting when \code{tilt} is \code{TRUE}.  If \code{tilt} is \code{TRUE} and 
they are not supplied then \code{tilt.boot} uses \code{empinf} to calculate them.

\item[\code{theta}] The required parameter value(s) for the tilted distribution(s).  There should be
one value of \code{theta} for each of the non-uniform distributions.  If \code{R[1]}
is 0 \code{theta} is a required argument.  Otherwise \code{theta} values can be estimated 
from the initial uniform bootstrap and the values in \code{alpha}.

\item[\code{alpha}] The alpha level to which tilting is required.  This parameter is ignored if
\code{R[1]} is 0 or if \code{theta} is supplied, otherwise it is used to find the values 
of \code{theta} as quantiles of the initial uniform bootstrap.  In this case \code{R[1]} 
should  be large enough that \code{min(c(alpha, 1-alpha))*R[1] > 5}, if this is not 
the case then a warning is generated to the effect that the \code{theta} are extreme 
values and so the tilted output may be unreliable.  

\item[\code{tilt}] A logical variable which if \code{TRUE} (the default) indicates that exponential 
tilting should be used, otherwise local frequency smoothing (\code{smooth.f}) is 
used.  If \code{tilt} is \code{FALSE} then \code{R[1]} must be positive.  In fact in this case 
the value of \code{R[1]} should be fairly large (in the region of 500 or more).

\item[\code{width}] This argument is used only if \code{tilt} is \code{FALSE}, in which case it is passed 
unchanged to \code{smooth.f} as the standardized bandwidth for the smoothing 
operation.  The value should generally be in the range (0.2, 1). See \code{smooth.f}
for for more details.

\item[\code{index}] The index of the statistic of interest in the output from \code{statistic}.  By 
default the first element of the output of \code{statistic} is used.

\item[\code{...}] Any additional arguments required by \code{statistic}.  These are passed unchanged to
\code{statistic} each time it is called.

\end{ldescription}
\end{Arguments}
\begin{Value}
An object of class \code{"boot"} with the following components

\begin{ldescription}
\item[\code{t0}] The observed value of the statistic on the original data.

\item[\code{t}] The values of the bootstrap replicates of the statistic.  There will be \code{sum(R)}
of these, the first \code{R[1]} corresponding to the uniform bootstrap and the 
remainder to the tilted bootstrap(s).

\item[\code{R}] The input vector of the number of bootstrap replicates.

\item[\code{data}] The original data as supplied.

\item[\code{statistic}] The \code{statistic} function as supplied.

\item[\code{sim}] The simulation type used in the bootstrap(s), it can either be \code{"ordinary"} or
\code{"balanced"}.

\item[\code{stype}] The type of statistic supplied, it is the same as the input value \code{stype}.

\item[\code{call}] A copy of the original call to \code{tilt.boot}.

\item[\code{strata}] The strata as supplied.

\item[\code{weights}] The matrix of weights used.  If \code{R[1]} is greater than 0 then the first row 
will be the uniform weights and each subsequent row the tilted weights.  
If \code{R[1]} equals 0 then the uniform
weights are omitted and only the tilted weights are output.

\item[\code{theta}] The values of \code{theta} used for the tilted distributions.  These are either the
input values or the values derived from the uniform bootstrap and \code{alpha}.

\end{ldescription}
\end{Value}
\begin{References}\relax
Booth, J.G., Hall, P. and Wood, A.T.A. (1993) Balanced importance resampling 
for the bootstrap. \emph{Annals of Statistics}, \bold{21}, 286--298.


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


Hinkley, D.V. and Shi, S. (1989) Importance sampling and the nested bootstrap. 
\emph{Biometrika}, \bold{76}, 435--446.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{boot}{boot}}, \code{\LinkA{exp.tilt}{exp.tilt}}, \code{\LinkA{Imp.Estimates}{Imp.Estimates}}, \code{\LinkA{imp.weights}{imp.weights}}, \code{\LinkA{smooth.f}{smooth.f}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
# Note that these examples can take a while to run.

# Example 9.9 of Davison and Hinkley (1997).
grav1 <- gravity[as.numeric(gravity[,2])>=7,]
grav.fun <- function(dat, w, orig)
{    strata <- tapply(dat[, 2], as.numeric(dat[, 2]))
     d <- dat[, 1]
     ns <- tabulate(strata)
     w <- w/tapply(w, strata, sum)[strata]
     mns <- tapply(d * w, strata, sum)
     mn2 <- tapply(d * d * w, strata, sum)
     s2hat <- sum((mn2 - mns^2)/ns)
     c(mns[2]-mns[1],s2hat,(mns[2]-mns[1]-orig)/sqrt(s2hat))
}
grav.z0 <- grav.fun(grav1,rep(1,26),0)
tilt.boot(grav1, grav.fun, R=c(249,375,375), stype="w", 
          strata=grav1[,2], tilt=TRUE, index=3, orig=grav.z0[1]) 

#  Example 9.10 of Davison and Hinkley (1997) requires a balanced 
#  importance resampling bootstrap to be run.  In this example we 
#  show how this might be run.  
acme.fun <- function(data, i, bhat)
{    d <- data[i,]
     n <- nrow(d)
     d.lm <- glm(d$acme~d$market)
     beta.b <- coef(d.lm)[2]
     d.diag <- glm.diag(d.lm)
     SSx <- (n-1)*var(d$market)
     tmp <- (d$market-mean(d$market))*d.diag$res*d.diag$sd
     sr <- sqrt(sum(tmp^2))/SSx
     c(beta.b, sr, (beta.b-bhat)/sr)
}
acme.b <- acme.fun(acme,1:nrow(acme),0)
acme.boot1 <- tilt.boot(acme, acme.fun, R=c(499, 250, 250), 
                        stype="i", sim="balanced", alpha=c(0.05, 0.95), 
                        tilt=TRUE, index=3, bhat=acme.b[1])
\end{ExampleCode}
\end{Examples}

