saddle.distn {boot}

Saddlepoint Distribution Approximations for Bootstrap Statistics

Description

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.

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

#  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)

# 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)