imp.weights               package:boot               R Documentation

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. 

       q: 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. 

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'.

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'.

Note:

     See the example in the help for 'imp.moments' for an example of
     using  'imp.weights'.

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.

See Also:

     'boot', 'exp.tilt', 'imp.moments', 'smooth.f', 'tilt.boot'

