Importance resampling using Chi-Square Tilting

Significant reduction in the value of the number of bootstrap resamples may be achieved when exponential tilting is used to estimate a tail probability or a quantile, associated with a small probability, say p. However, exponential tilting is based on the assumption that the bootstrap distribution is approximately normal. In this paper, chisquare tilting is introduced when the bootstrap distribution of the statistics of interest is assumed to have a skewed distribution. In the case of small value of the new tilting reduces the asymptotic variance by at least a factor of 6, implies a large saving in bootstrap resamples. Calculations show that the variance reduction is very close to the asymptotic one even for sample sizes as small as 15. Some asymptotic results are given. Computer simulation as well as real data is used to illustrate the computation for the lower bootstrap confidence limit of the bootstrap sample variance. Although, bootstrap confidence intervals for a variance are known to perform very badly in practice even when sampling from a normal distribution, see Schenker (1987), our simulation indicates that our method perform reasonably well.