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.

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, …)

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.

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

Aliases

• saddle.distn

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)
# }