imp.weights: Importance Sampling Weights

Description

This function calculates the importance sampling weight required to correct for simulation from a distribution with probabilities p when estimates are required assuming that simulation was from an alternative distribution with probabilities q.

Usage

imp.weights(boot.out, def = TRUE, q = NULL)

Arguments

boot.out

A object of class "boot" generated by boot or tilt.boot. Typically the bootstrap simulations would have been done using importance resampling and we wish to do our calculations under the assumption of sampling with equal probabilities.

def

A logical variable indicating whether the defensive mixture distribution weights should be calculated. This makes sense only in the case where the replicates in boot.out were simulated under a number of different distributions. If this is the case then the defensive mixture weights use a mixture of the distributions used in the bootstrap. The alternative is to calculate the weights for each replicate using knowledge of the distribution from which the bootstrap resample was generated.

A vector of probabilities specifying the resampling distribution from which we require inferences to be made. In general this would correspond to the usual bootstrap resampling distribution which gives equal weight to each of the original observations and this is the default. q must have length equal to the number of observations in the boot.out$data and all elements of q must be positive.

Value

A vector of importance weights of the same length as boot.out$t. These weights can then be used to reweight boot.out$t so that estimates can be found as if the simulations were from a distribution with probabilities q.

Details

The importance sampling weight for a bootstrap replicate with frequency vector f is given by prod((q/p)^f). This reweights the replicates so that estimates can be found as if the bootstrap resamples were generated according to the probabilities q even though, in fact, they came from the distribution p.

References

Davison, A. C. and Hinkley, D. V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

Hesterberg, T. (1995) Weighted average importance sampling and defensive mixture distributions. Technometrics, 37, 185--194.

Johns, M.V. (1988) Importance sampling for bootstrap confidence intervals. Journal of the American Statistical Association, 83, 709--714.