# saddle.distn

##### Saddlepoint Distribution Approximations for Bootstrap Statistics

Approximate an entire distribution using saddlepoint methods. This
function can calculate simple and conditional saddlepoint distribution
approximations for a univariate quantity of interest. For the simple
saddlepoint the quantity of interest is a linear combination of
**W** where **W** is a vector of random variables. For the
conditional saddlepoint we require the distribution of one linear
combination given the values of any number of other linear
combinations. The distribution of **W** must be one of multinomial,
Poisson or binary. The primary use of this function is to calculate
quantiles of bootstrap distributions using saddlepoint approximations.
Such quantiles are required by the function `control`

to
approximate the distribution of the linear approximation to a
statistic.

- Keywords
- smooth, dplot, nonparametric

##### Usage

```
saddle.distn(A, u = NULL, alpha = NULL, wdist = "m",
type = "simp", npts = 20, t = NULL, t0 = NULL,
init = rep(0.1, d), mu = rep(0.5, n), LR = FALSE,
strata = NULL, …)
```

##### Arguments

- A
This is a matrix of known coefficients or a function which returns such a matrix. If a function then its first argument must be the point

`t`

at which a saddlepoint is required. The most common reason for A being a function would be if the statistic is not itself a linear combination of the**W**but is the solution to a linear estimating equation.- u
If

`A`

is a function then`u`

must also be a function returning a vector with length equal to the number of columns of the matrix returned by`A`

. Usually all components other than the first will be constants as the other components are the values of the conditioning variables. If`A`

is a matrix with more than one column (such as when`wdist = "cond"`

) then`u`

should be a vector with length one less than`ncol(A)`

. In this case`u`

specifies the values of the conditioning variables. If`A`

is a matrix with one column or a vector then`u`

is not used.- alpha
The alpha levels for the quantiles of the distribution which should be returned. By default the 0.1, 0.5, 1, 2.5, 5, 10, 20, 50, 80, 90, 95, 97.5, 99, 99.5 and 99.9 percentiles are calculated.

- wdist
The distribution of

**W**. Possible values are`"m"`

(multinomial),`"p"`

(Poisson), or`"b"`

(binary).- type
The type of saddlepoint to be used. Possible values are

`"simp"`

(simple saddlepoint) and`"cond"`

(conditional). If`wdist`

is`"m"`

,`type`

is set to`"simp"`

.- npts
The number of points at which the saddlepoint approximation should be calculated and then used to fit the spline.

- t
A vector of points at which the saddlepoint approximations are calculated. These points should extend beyond the extreme quantiles required but still be in the possible range of the bootstrap distribution. The observed value of the statistic should not be included in

`t`

as the distribution function approximation breaks down at that point. The points should, however cover the entire effective range of the distribution including close to the centre. If`t`

is supplied then`npts`

is set to`length(t)`

. When`t`

is not supplied, the function attempts to find the effective range of the distribution and then selects points to cover this range.- t0
If

`t`

is not supplied then a vector of length 2 should be passed as`t0`

. The first component of`t0`

should be the centre of the distribution and the second should be an estimate of spread (such as a standard error). These two are then used to find the effective range of the distribution. The range finding mechanism does rely on an accurate estimate of location in`t0[1]`

.- init
When

`wdist`

is`"m"`

, this vector should contain the initial values to be passed to`nlmin`

when it is called to solve the saddlepoint equations.- mu
The vector of parameter values for the distribution. The default is that the components of

**W**are identically distributed.- LR
A logical flag. When

`LR`

is`TRUE`

the Lugananni-Rice cdf approximations are calculated and used to fit the spline. Otherwise the cdf approximations used are based on Barndorff-Nielsen's r*.- strata
A vector giving the strata when the rows of A relate to stratified data. This is used only when

`wdist`

is`"m"`

.- …
When

`A`

and`u`

are functions any additional arguments are passed unchanged each time one of them is called.

##### Details

The range at which the saddlepoint is used is such that the cdf
approximation at the endpoints is more extreme than required by the
extreme values of `alpha`

. The lower endpoint is found by
evaluating the saddlepoint at the points `t0[1]-2*t0[2]`

,
`t0[1]-4*t0[2]`

, `t0[1]-8*t0[2]`

etc. until a point is
found with a cdf approximation less than `min(alpha)/10`

, then a
bisection method is used to find the endpoint which has cdf
approximation in the range (`min(alpha)/1000`

,
`min(alpha)/10`

). Then a number of, equally spaced, points are
chosen between the lower endpoint and `t0[1]`

until a total of
`npts/2`

approximations have been made. The remaining
`npts/2`

points are chosen to the right of `t0[1]`

in a
similar manner. Any points which are very close to the centre of the
distribution are then omitted as the cdf approximations are not
reliable at the centre. A smoothing spline is then fitted to the
probit of the saddlepoint distribution function approximations at the
remaining points and the required quantiles are predicted from the
spline.

Sometimes the function will terminate with the message
`"Unable to find range"`

. There are two main reasons why this may
occur. One is that the distribution is too discrete and/or the
required quantiles too extreme, this can cause the function to be
unable to find a point within the allowable range which is beyond the
extreme quantiles. Another possibility is that the value of
`t0[2]`

is too small and so too many steps are required to find
the range. The first problem cannot be solved except by asking for
less extreme quantiles, although for very discrete distributions the
approximations may not be very good. In the second case using a
larger value of `t0[2]`

will usually solve the problem.

##### Value

The returned value is an object of class `"saddle.distn"`

. See the help
file for `saddle.distn.object`

for a description of such
an object.

##### 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.

##### See Also

`lines.saddle.distn`

, `saddle`

,
`saddle.distn.object`

, `smooth.spline`

##### Examples

```
# NOT RUN {
# The bootstrap distribution of the mean of the air-conditioning
# failure data: fails to find value on R (and probably on S too)
air.t0 <- c(mean(aircondit$hours), sqrt(var(aircondit$hours)/12))
# }
# NOT RUN {
saddle.distn(A = aircondit$hours/12, t0 = air.t0)
# }
# NOT RUN {
# alternatively using the conditional poisson
saddle.distn(A = cbind(aircondit$hours/12, 1), u = 12, wdist = "p",
type = "cond", t0 = air.t0)
# Distribution of the ratio of a sample of size 10 from the bigcity
# data, taken from Example 9.16 of Davison and Hinkley (1997).
ratio <- function(d, w) sum(d$x *w)/sum(d$u * w)
city.v <- var.linear(empinf(data = city, statistic = ratio))
bigcity.t0 <- c(mean(bigcity$x)/mean(bigcity$u), sqrt(city.v))
Afn <- function(t, data) cbind(data$x - t*data$u, 1)
ufn <- function(t, data) c(0,10)
saddle.distn(A = Afn, u = ufn, wdist = "b", type = "cond",
t0 = bigcity.t0, data = bigcity)
# From Example 9.16 of Davison and Hinkley (1997) again, we find the
# conditional distribution of the ratio given the sum of city$u.
Afn <- function(t, data) cbind(data$x-t*data$u, data$u, 1)
ufn <- function(t, data) c(0, sum(data$u), 10)
city.t0 <- c(mean(city$x)/mean(city$u), sqrt(city.v))
saddle.distn(A = Afn, u = ufn, wdist = "p", type = "cond", t0 = city.t0,
data = city)
# }
```

*Documentation reproduced from package boot, version 1.3-24, License: Unlimited*