Exponential Tilting

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)))

Arguments

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.

Value

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.

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.

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$p,
     strata = grav1[,2], orig = grav.z0[1])
#> 
#> STRATIFIED WEIGHTED BOOTSTRAP
#> 
#> 
#> Call:
#> boot(data = grav1, statistic = grav.fun, R = 499, stype = "w", 
#>     strata = grav1[, 2], weights = grav.tilt$p, orig = grav.z0[1])
#> 
#> 
#> Bootstrap Statistics :
#>     original     bias    std. error  mean(t*)
#> t1* 2.846154 -0.0683528   1.4324110  5.591182
#> t2* 2.392353 -0.2301600   0.9030430  3.331344
#> t3* 0.000000 -0.1710658   0.9713051  1.430849