Efficiency balanced designs for bootstrap simulations

This paper considers the balanced resampling design approach of generating bootstrap samples to handle the simulation error problem in bootstrap estimation of mean and variance of a sample statistic of interest. A class of balanced n-arry block design, slightly different from the definition of Tocher (1952), is defined and its necessary conditions for the existence of these designs are derived. In particular, an infinite series of balanced n-arry block designs are found to be efficiency balanced, i.e., every contrast of element (observation) totals has the same loss of information. As an application in bootstrap method, the eccentric class of efficiency balanced n-arry block design (termed Efficient Bootstrap Sample Design (EBSD(n)) can be used to generate 4n2+1 resamples of sample size n to achieve an exact bootstrap estimation of mean and variance of statistic “sample average”, i.e., a collection of bootstrap data samples and corresponding bootstrap sample averages where mean and variance of this empirical collection of sample averages match the first two theoretical central moments i.e., moments about the sample average. The EBSD(n) solely depends on the existence of balanced incomplete block design B[2n, 4n-2, 2n-1, n, n-1] for n ≥ 2. Thus the implementation of computer algorithm for spawning bootstrap samples using EBSD(n) is simple and straightforward for recapitulating sampling characteristics such as standard error of predefined computable symmetric and non symmetric sample statistic of interest.