| exp.tilt {boot} | R Documentation |
This function calculates exponentially tilted multinomial distributions such that the resampling distributions of the linear approximation to a statistic have the required means.
exp.tilt(L, theta=NULL, t0=0, lambda=NULL,
strata=rep(1, length(L)))
L |
The empirical influence values for the statistic of interest based on the
observed data. The length of L should be the same as the size of the
original data set. Typically L will be calculated by a call to empinf.
|
theta |
The value at which the tilted distribution is to be centred. This is not
required if lambda is supplied but is needed otherwise.
|
t0 |
The current value of the statistic. The default is that the statistic equals 0. |
lambda |
The Lagrange multiplier(s). For each value of lambda a multinomial
distribution
is found with probabilities proportional to exp(lambda * L). In general
lambda is not known and so theta would be supplied, and the corresponding
value of lambda found. If both lambda and theta are supplied then
lambda is ignored and the multipliers for tilting to theta are found.
|
strata |
A vector or factor of the same length as L giving the strata for the
observed data and the empirical influence values L.
|
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 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.
A list with the following components :
p |
The tilted probabilities. There will be m distributions where m is the
length of theta (or lambda if supplied). If m is 1 then p is a vector
of length(L) probabilities. If m is greater than 1 then p is a matrix
with m rows, each of which contain length(L) probabilities. In this case
the vector p[i,] is the distribution tilted to theta[i]. p is
in the form required by the argument weights of the function boot for
importance resampling.
|
lambda |
The Lagrange multiplier used in the equation to determine the tilted
probabilities. lambda is a vector of the same length as theta.
|
theta |
The values of theta to which the distributions have been tilted. In general
this will be the input value of theta but if lambda was supplied then
this is the vector of the corresponding theta values.
|
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.
empinf, imp.prob, imp.quantile, optim, smooth.f, tilt.boot
# 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 <- tapply(d * w, strata, sum)
mn2 <- tapply(d * d * w, strata, sum)
s2hat <- sum((mn2 - mns^2)/ns)
as.vector(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])