Abstract: For constructing inferential procedures from a random sample of size n, Graham et al. (1990) proposed second order balanced bootstrap designs that reduce the variance in the usual, unbalanced bootstrap simulation. Their methods, however, do not cover the following cases: (i) n, a composite odd number; (ii) n=4m+1, a prime number. Here we first give two methods that provide second-order balanced designs for all cases. We then extend the results to stratified multistage samples, and construct balanced bootstrap designs, for the important special case of equal first-stage sample sizes within strata, yielding second-order balance.
Key words and phrases: Balanced incomplete block designs, bootstrap, Hadamard matrices, quadratic residues, second-order balance, stratified sampling.