THE LIMITING DISTRIBUTION OF A RECURSIVE RESAMPLING PROCEDURE

A recursive resampling method is discussed in this paper. Let X(1), X( 2), ..., X(n) be i.i.d. random variables with distribution function F and construct the empirical distribution function F-n. A new sample X( n+1) is drawn from F-n and the new empirical distribution function ($) over tilde F-n+1 in the wide sense, is computed from X(1), X(2), ..., X(n), X(n+1). Then X(n+2) is drawn from ($) over tilde F-n+1 and ($) over tilde F-n+2 is obtained. In this way, X(n+m) and ($) over tilde F -n+m are found. It will be proved that ($) over tilde F-n+m converges to a random variable almost surely as m goes to infinity and the limit ing distribution is a compound beta distribution. In comparison with t he usual non-recursive bootstrap, the main advantage of this procedure is a reduction in unconditional variance.