\HeaderA{saddle}{Saddlepoint Approximations for Bootstrap Statistics}{saddle}
\keyword{smooth}{saddle}
\keyword{nonparametric}{saddle}
\begin{Description}\relax
This function calculates a saddlepoint approximation to the distribution of
a linear combination of \bold{W} at a particular point \code{u}, where \bold{W}
is a vector of random variables.  The distribution 
of \bold{W} may be multinomial (default), Poisson or binary.  Other
distributions are possible also if the 
adjusted cumulant generating function and its second derivative are given.  
Conditional saddlepoint approximations to the distribution of one linear 
combination given the values of
other linear combinations of \bold{W} can be calculated for \bold{W} having
binary or Poisson distributions.
\end{Description}
\begin{Usage}
\begin{verbatim}
saddle(A=NULL, u=NULL, wdist="m", type="simp", d=NULL, d1=1, 
       init=rep(0.1, d), mu=rep(0.5, n), LR=FALSE, strata=NULL, 
       K.adj=NULL, K2=NULL)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{A}] A vector or matrix of known coefficients of the linear combinations of \bold{W}.
It is a required argument unless \code{K.adj} and \code{K2} are supplied, in which case 
it is ignored.

\item[\code{u}] The value at which it is desired to calculate the saddlepoint approximation to
the distribution of the linear combination of \bold{W}. It is a required
argument unless \code{K.adj} and \code{K2} are supplied, in which case it is ignored.

\item[\code{wdist}] The distribution of \bold{W}.  This can be one of \code{"m"} (multinomial), \code{"p"} 
(Poisson), \code{"b"} (binary) or "o" (other).  If K.adj and K2 are given  
\code{wdist} is set to "o".

\item[\code{type}] The type of saddlepoint approximation.  Possible types are \code{"simp"} for simple
saddlepoint and \code{"cond"} for the conditional saddlepoint.  When \code{wdist} is
\code{"o"} or \code{"m"}, \code{type} is automatically set to \code{"simp"}, which is the only
type of saddlepoint currently implemented for those distributions.

\item[\code{d}] This specifies the dimension of the whole statistic.  This argument is required
only when \code{wdist="o"} and defaults to 1 if not supplied in that case.  For
other distributions it is set to \code{ncol(A)}.

\item[\code{d1}] When \code{type} is \code{"cond"} this is the dimension of the statistic of interest
which must be less than \code{length(u)}.  Then the saddlepoint approximation to the
conditional distribution of the first \code{d1} linear combinations given the values
of the remaining combinations is found.  Conditional distribution function
approximations can only be found if the value of \code{d1} is 1. 

\item[\code{init}] Used if \code{wdist} is either \code{"m"} or \code{"o"}, this gives initial values to 
\code{nlmin} which is used to solve the saddlepoint equation.  

\item[\code{mu}] The values of the parameters of the distribution of \bold{W} when \code{wdist} is
\code{"m"}, \code{"p"} \code{"b"}.  \code{mu} must be of the same length as W (i.e. \code{nrow(A)}).
The default is that all values of \code{mu} are equal and so the elements of \bold{W}
are identically distributed.

\item[\code{LR}] If \code{TRUE} then the Lugananni-Rice approximation to the cdf is used,
otherwise the approximation used is based on Barndorff-Nielsen's r*. 

\item[\code{strata}] The strata for stratified data.

\item[\code{K.adj}] The adjusted cumulant generating function used when \code{wdist} is \code{"o"}.  
This is a function of a single parameter, \code{zeta},  which calculates 
\code{K(zeta)-u\%*\%zeta}, where \code{K(zeta)} is the cumulant generating function 
of \bold{W}.

\item[\code{K2}] This is a function of a single parameter \code{zeta} which returns the 
matrix of second derivatives of \code{K(zeta)} for use when \code{wdist} is \code{"o"}.  
If \code{K.adj} is given then this must be given also.  It is called only once
with the calculated solution to the saddlepoint equation being passed as 
the argument.  This argument is ignored if \code{K.adj} is not supplied.

\end{ldescription}
\end{Arguments}
\begin{Details}\relax
If \code{wdist} is \code{"o"} or \code{"m"}, the saddlepoint equations are solved using
\code{nlmin} to minimize \code{K.adj} with respect to its parameter \code{zeta}. 
For the Poisson and binary cases, a generalized linear model is fitted such 
that the parameter 
estimates solve the saddlepoint equations.  The response variable 'y' for the
\code{glm} must satisfy the equation \code{t(A)\%*\%y=u} (\code{t()} being the transpose 
function).  Such a vector can be found
as a feasible solution to a linear programming problem.  This is done
by a call to \code{simplex}.  The covariate matrix for the \code{glm} is given by \code{A}.
\end{Details}
\begin{Value}
A list consisting of the following components

\begin{ldescription}
\item[\code{spa}] The saddlepoint approximations.  The first value is the density approximation
and the second value is the distribution function approximation.

\item[\code{zeta.hat}] The solution to the saddlepoint equation.  For the conditional saddlepoint
this is the solution to the saddlepoint equation for the numerator.

\item[\code{zeta2.hat}] If \code{type} is \code{"cond"} this is the solution to the saddlepoint equation for the 
denominator.  This component is not returned for any other value of \code{type}.

\end{ldescription}
\end{Value}
\begin{References}\relax
Booth, J.G. and Butler, R.W. (1990) Randomization distributions and 
saddlepoint approximations in generalized linear models. 
\emph{Biometrika}, \bold{77}, 787--796.


Canty, A.J. and Davison, A.C. (1997) Implementation of saddlepoint 
approximations to resampling distributions.  
\emph{Computing Science and Statistics; Proceedings of the 28th Symposium on the Interface}, 248--253.


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


Jensen, J.L. (1995) \emph{Saddlepoint Approximations}. Oxford University Press.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{saddle.distn}{saddle.distn}}, \code{\LinkA{simplex}{simplex}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
# To evaluate the bootstrap distribution of the mean failure time of 
# air-conditioning equipment at 80 hours
saddle(A=aircondit$hours/12, u=80)

# Alternatively this can be done using a conditional poisson
saddle(A=cbind(aircondit$hours/12,1), u=c(80,12), wdist="p", type="cond")

# To use the Lugananni-Rice approximation to this
saddle(A=cbind(aircondit$hours/12,1), u=c(80,12), wdist="p", type="cond", 
       LR = TRUE)

# Example 9.16 of Davison and Hinkley (1997) calculates saddlepoint 
# approximations to the distribution of the ratio statistic for the
# city data. Since the statistic is not in itself a linear combination
# of random Variables, its distribution cannot be found directly.  
# Instead the statistic is expressed as the solution to a linear 
# estimating equation and hence its distribution can be found.  We
# get the saddlepoint approximation to the pdf and cdf evaluated at
# t=1.25 as follows.
jacobian <- function(dat,t,zeta)
{
     p <- exp(zeta*(dat$x-t*dat$u))
     abs(sum(dat$u*p)/sum(p))
}
city.sp1 <- saddle(A=city$x-1.25*city$u, u=0)
city.sp1$spa[1] <- jacobian(city, 1.25, city.sp1$zeta.hat) * city.sp1$spa[1]
city.sp1
\end{ExampleCode}
\end{Examples}

