Saddlepoint Approximations for Bootstrap Statistics
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 |
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 |
wdist |
The distribution of W. This can be one of |
type |
The type of saddlepoint approximation. Possible types are
|
d |
This specifies the dimension of the whole statistic. This argument
is required only when |
d1 |
When |
init |
Used if |
mu |
The values of the parameters of the distribution of W when
|
LR |
If |
strata |
The strata for stratified data. |
K.adj |
The adjusted cumulant generating function used when |
K2 |
This is a function of a single parameter |
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 |
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.sp1Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.