## Saddlepoint Approximations for Bootstrap Statistics

### Description

This function calculates a saddlepoint approximation to the distribution of a linear combination of W at a particular point `u`, where W is a vector of random variables. The distribution of 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 W can be calculated for W having binary or Poisson distributions.

### Usage

```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,
```

### Arguments

 `A` A vector or matrix of known coefficients of the linear combinations of W. It is a required argument unless `K.adj` and `K2` are supplied, in which case it is ignored. `u` The value at which it is desired to calculate the saddlepoint approximation to the distribution of the linear combination of W. It is a required argument unless `K.adj` and `K2` are supplied, in which case it is ignored. `wdist` The distribution of W. This can be one of `"m"` (multinomial), `"p"` (Poisson), `"b"` (binary) or "o" (other). If K.adj and K2 are given `wdist` is set to "o". `type` The type of saddlepoint approximation. Possible types are `"simp"` for simple saddlepoint and `"cond"` for the conditional saddlepoint. When `wdist` is `"o"` or `"m"`, `type` is automatically set to `"simp"`, which is the only type of saddlepoint currently implemented for those distributions. `d` This specifies the dimension of the whole statistic. This argument is required only when `wdist="o"` and defaults to 1 if not supplied in that case. For other distributions it is set to `ncol(A)`. `d1` When `type` is `"cond"` this is the dimension of the statistic of interest which must be less than `length(u)`. Then the saddlepoint approximation to the conditional distribution of the first `d1` linear combinations given the values of the remaining combinations is found. Conditional distribution function approximations can only be found if the value of `d1` is 1. `init` Used if `wdist` is either `"m"` or `"o"`, this gives initial values to `nlmin` which is used to solve the saddlepoint equation. `mu` The values of the parameters of the distribution of W when `wdist` is `"m"`, `"p"` `"b"`. `mu` must be of the same length as W (i.e. `nrow(A)`). The default is that all values of `mu` are equal and so the elements of W are identically distributed. `LR` If `TRUE` then the Lugananni-Rice approximation to the cdf is used, otherwise the approximation used is based on Barndorff-Nielsen's r*. `strata` The strata for stratified data. `K.adj` The adjusted cumulant generating function used when `wdist` is `"o"`. This is a function of a single parameter, `zeta`, which calculates `K(zeta)-u%*%zeta`, where `K(zeta)` is the cumulant generating function of W. `K2` This is a function of a single parameter `zeta` which returns the matrix of second derivatives of `K(zeta)` for use when `wdist` is `"o"`. If `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 `K.adj` is not supplied.

### Details

If `wdist` is `"o"` or `"m"`, the saddlepoint equations are solved using `nlmin` to minimize `K.adj` with respect to its parameter `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 `glm` must satisfy the equation `t(A)%*%y=u` (`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 `simplex`. The covariate matrix for the `glm` is given by `A`.

### Value

A list consisting of the following components

 `spa` The saddlepoint approximations. The first value is the density approximation and the second value is the distribution function approximation. `zeta.hat` The solution to the saddlepoint equation. For the conditional saddlepoint this is the solution to the saddlepoint equation for the numerator. `zeta2.hat` If `type` is `"cond"` this is the solution to the saddlepoint equation for the denominator. This component is not returned for any other value of `type`.

### References

Booth, J.G. and Butler, R.W. (1990) Randomization distributions and saddlepoint approximations in generalized linear models. Biometrika, 77, 787–796.

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

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

Jensen, J.L. (1995) Saddlepoint Approximations. Oxford University Press.

`saddle.distn`, `simplex`

### Examples

```# To evaluate the bootstrap distribution of the mean failure time of
# air-conditioning equipment at 80 hours

# Alternatively this can be done using a conditional poisson

# To use the Lugananni-Rice approximation to this
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))
}