exp.tilt
Exponential Tilting
Description
This function calculates exponentially tilted multinomial distributions such that the resampling distributions of the linear approximation to a statistic have the required means.
Usage
exp.tilt(L, theta = NULL, t0 = 0, lambda = NULL, strata = rep(1, length(L)))
Arguments
L | The empirical influence values for the statistic of interest based on the observed data. The length of |
theta | The value at which the tilted distribution is to be centred. This is not required if |
t0 | The current value of the statistic. The default is that the statistic equals 0. |
lambda | The Lagrange multiplier(s). For each value of |
strata | A vector or factor of the same length as |
Details
Exponential tilting involves finding a set of weights for a data set to ensure that the bootstrap distribution of the linear approximation to a statistic of interest has mean theta
. The weights chosen to achieve this are given by p[j]
proportional to exp(lambda*L[j]/n)
, where n
is the number of data points. lambda
is then chosen to make the mean of the bootstrap distribution, of the linear approximation to the statistic of interest, equal to the required value theta
. Thus lambda
is defined as the solution of a nonlinear equation. The equation is solved by minimizing the Euclidean distance between the left and right hand sides of the equation using the function nlmin
. If this minimum is not equal to zero then the method fails.
Typically exponential tilting is used to find suitable weights for importance resampling. If a small tail probability or quantile of the distribution of the statistic of interest is required then a more efficient simulation is to centre the resampling distribution close to the point of interest and then use the functions imp.prob
or imp.quantile
to estimate the required quantity.
Another method of achieving a similar shifting of the distribution is through the use of smooth.f
. The function tilt.boot
uses exp.tilt
or smooth.f
to find the weights for a tilted bootstrap.
Value
A list with the following components :
p | The tilted probabilities. There will be |
lambda | The Lagrange multiplier used in the equation to determine the tilted probabilities. |
theta | The values of |
References
Davison, A. C. and Hinkley, D. V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.
Efron, B. (1981) Nonparametric standard errors and confidence intervals (with Discussion). Canadian Journal of Statistics, 9, 139–172.
See Also
empinf
, imp.prob
, imp.quantile
, optim
, smooth.f
, tilt.boot
Examples
# Example 9.8 of Davison and Hinkley (1997) requires tilting the resampling # distribution of the studentized statistic to be centred at the observed # value of the test statistic 1.84. This can be achieved as follows. grav1 <- gravity[as.numeric(gravity[,2]) >=7 , ] grav.fun <- function(dat, w, orig) { strata <- tapply(dat[, 2], as.numeric(dat[, 2])) d <- dat[, 1] ns <- tabulate(strata) w <- w/tapply(w, strata, sum)[strata] mns <- as.vector(tapply(d * w, strata, sum)) # drop names mn2 <- tapply(d * d * w, strata, sum) s2hat <- sum((mn2 - mns^2)/ns) c(mns[2]-mns[1], s2hat, (mns[2]-mns[1]-orig)/sqrt(s2hat)) } grav.z0 <- grav.fun(grav1, rep(1, 26), 0) grav.L <- empinf(data = grav1, statistic = grav.fun, stype = "w", strata = grav1[,2], index = 3, orig = grav.z0[1]) grav.tilt <- exp.tilt(grav.L, grav.z0[3], strata = grav1[,2]) boot(grav1, grav.fun, R = 499, stype = "w", weights = grav.tilt$p, strata = grav1[,2], orig = grav.z0[1])
Copyright (©) 1999–2012 R Foundation for Statistical Computing.
Licensed under the GNU General Public License.