Imp.Estimates              package:boot              R Documentation

Importance Sampling Estimates

Description:

     Central moment, tail probability, and quantile estimates for a
     statistic under importance resampling.

Usage:

     imp.moments(boot.out=NULL, index=1, t=boot.out$t[, index], 
              w=NULL, def=TRUE, q=NULL)
     imp.prob(boot.out=NULL, index=1, t0=boot.out$t0[index], 
              t=boot.out$t[, index], w=NULL, def=TRUE, q=NULL)
     imp.quantile(boot.out=NULL, alpha=NULL, index=1, 
              t=boot.out$t[, index], w=NULL, def=TRUE, q=NULL)

Arguments:

boot.out: A object of class '"boot"' generated by a call to 'boot' or
          'tilt.boot'.  Use of these functions makes sense only when
          the bootstrap resampling used unequal weights for the
          observations.  If the importance weights 'w' are not 
          supplied then 'boot.out' is a required argument. It is also
          required if 't' is not supplied. 

   alpha: The alpha levels for the required quantiles.  The default is
          to calculate the 1%, 2.5%, 5%, 10%, 90%, 95%, 97.5% and 99%
          quantiles. 

   index: The index of the variable of interest in the output of
          'boot.out$statistic'. This is not used if the argument 't' is
          supplied. 

      t0: The values at which tail probability estimates are required. 
          For each value 't0[i]' the function will estimate the
          bootstrap cdf evaluated at 't0[i]'.  If 'imp.prob' is called
          without the argument 't0' then the bootstrap cdf evaluated at
          the observed value of the statistic is found. 

       t: The bootstrap replicates of a statistic.  By default these
          are taken from the bootstrap output object 'boot.out' but
          they can be supplied separately if required (e.g. when the
          statistic of interest is a function of the calculated values
          in 'boot.out').  Either 'boot.out' or 't' must be supplied. 

       w: The importance resampling weights for the bootstrap
          replicates.  If they are not supplied then 'boot.out' must be
          supplied, in which case the importance weights are calculated
          by a call to 'imp.weights'. 

     def: A logical value indicating whether a defensive mixture is to
          be used for weight calculation.  This is used only if 'w' is
          missing and it is passed unchanged  to 'imp.weights' to
          calculate 'w'. 

       q: A vector of probabilities specifying the resampling
          distribution from which  any estimates should be found.  In
          general this would correspond to the usual bootstrap
          resampling distribution which gives equal weight to each of
          the  original observations. The estimates depend on this
          distribution only through  the importance weights 'w' so this
          argument is ignored if 'w' is supplied.  If 'w' is missing
          then 'q' is passed as an argument to 'imp.weights' and used
          to find 'w'. 

Value:

     A list with the following components :

   alpha: The 'alpha' levels used for the quantiles, if 'imp.quantile'
          is used. 

      t0: The values at which the tail probabilities are estimated, if
          'imp.prob' is used. 

     raw: The raw importance resampling estimates.  For 'imp.moments'
          this has length 2, the first component being the estimate of
          the mean and the second being the variance estimate.  For
          'imp.prob', 'raw' is of the same length as 't0', and for
          'imp.quantile' it is of the same length as 'alpha'. 

     rat: The ratio importance resampling estimates.  In this method
          the weights 'w' are rescaled to have average value one before
          they are used.  The format of this vector is the same as
          'raw'. 

     reg: The regression importance resampling estimates.  In this
          method the weights which are used are derived from a
          regression of 't*w' on 'w'.  This choice of weights can be
          shown to minimize the variance of the weights and also the
          Euclidean distance of the weights from the uniform weights. 
          The format of this  vector is the same as 'raw'. 

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.weights', '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.  In this example
     # we show how certain estimates can be found using resamples taken from
     # the tilted distribution.
     data(gravity)
     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])
     grav.tilt.boot <- boot(grav1, grav.fun, R=199, stype="w", 
                            strata=grav1[,2], weights=grav.tilt$p,
                            orig=grav.z0[1])
     # Since the weights are needed for all calculations, we shall calculate
     # them once only.
     grav.w <- imp.weights(grav.tilt.boot)
     grav.mom <- imp.moments(grav.tilt.boot, w=grav.w, index=3)
     grav.p <- imp.prob(grav.tilt.boot, w=grav.w, index=3, t0=grav.z0[3])
     grav.q <- imp.quantile(grav.tilt.boot, w=grav.w, index=3, 
                            alpha=c(0.9,0.95,0.975,0.99))

