### Name: saddle
### Title: Saddlepoint Approximations for Bootstrap Statistics
### Aliases: saddle
### Keywords: smooth nonparametric

### ** Examples

# 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