Efficient construction of a smooth nonparametric family of empirical distributions and calculation of bootstrap likelihood

This paper considers the efficient construction of a nonparametric family of distributions indexed by a specified parameter of interest and its application to calculating a bootstrap likelihood for the parameter. An approximate expression is obtained for the variance of log bootstrap likelihood for statistics which are defined by an estimating equation resulting from the method of selecting the first-level bootstrap populations and parameters. The expression is shown to agree well with simulations for artificial data sets based on quantiles of the standard normal distribution, and these results give guidelines for the amount of aggregation of bootstrap samples with similar parameter values required to achieve a given reduction in variance. An application to earthquake data illustrates how the variance expression can be used to construct an efficient Monte Carlo algorithm for defining a smooth nonparametric family of empirical distributions to calculate a bootstrap likelihood by greatly reducing the inherent variability due to first-level resampling.