| saddle {boot} | R Documentation |
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.
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)
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.
|
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.
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.
|
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.
# 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