RESAMPLING STUDENT'S t-TYPE STATISTICS

The present paper establishes conditional and unconditional central limit theorems for various resampling procedures for the t-statistic. The results work under fairly general conditions and the underlying random variables need not to be independent. Specific examples are then the 're(n) (double) bootstrap out of k(n) observations, the Bayesian bootstrap and two-sample t-type permutation statistics. In case when m(n)/k(n) is bounded away from zero and infinity necessary and sufficient conditions for the conditional central limit law of the bootstrap t-statistics are established. For high resampling intensity when m(n)/k(n) tends to infinity the following general result is obtained. Without further other assumptions the bootstrap makes the resampled t-statistic automatically normal. The results are based on a general conditional limit theorem for weighted resampling statistics which is of own interest. Consider a triangular array of arbitrary real random variables Xn,i, 1 < i < k(n), on some probability space (gt, ,4, P) with k(n)-+ ~x~ as n-~ c~. The one-sample t-statistic is then (1.1) t~= k(~-l/2 ~-~k(n) Xn,i \'~1 A.~i-1 1 ~-]ki:(~)(Xni_-Rn)2) k(n)-1 = ' i/2 given by the mean X~ = 1/k(n) V'k(n) Xn ~. Throughout we will discuss the limit behaviour of various resampling versions of tn and tests of t-test type. Specific examples are all kind of bootstrap and permutation resampling statistics. The results are typically applied in testing statistical hypotheses when critical values of tn are derived by resampling methods under a nonparametric null hypothesis, see Section 5 and Janssen and Pauls (2003). At this stage we like to emphasize that the Xn# may come from arbitrary alternatives which may no longer be independent or identically distributed. In a forthcoming paper the present results will be applied to establish power functions for resampling tests. Two-sample t-statistics are treated similarly in Section 4. All proofs can be found in Section 6. In the next step let us draw re(n) resampling variables X~', , Xm n given the 9 .. () original data Xn#. Below various different resampling schemes are discussed in detail.