The Deficiency Introduced by Resampling

Abstract

When the classical nonparametric bootstrap is implemented by a Monte-Carlo procedure one resamples values from a sequence of, typically, independent and identically distributed ones. But what happens when a decision has to be taken based on such resampled values? One way to quantify the loss of information due to this resampling step is to consider the deficiency distance, in the sense of Le Cam, between a statistical experiment of n independent and identically distributed observations and the one consisting of m observations taken from the original n by resampling with replacement. By comparing with an experiment where only subsamplingwith a random subsampling size has been performed one can bound the deficiency in terms of the amount of information contained in additional observations. It follows for certain experiments that the deficiency distance is proportional to the expected fraction of observations missed when resampling.