ASYMPTOTIC COMPARISONS OF SEVERAL VARIANCE ESTIMATORS AND THEIR EFFECTS FOR STUDENTIZATIONS

YOSHIHIKO MAESONO

Centre for Mathematics and Its Applications, School of Mathematical Sciences, The Australian National University, Canberra, ACT 0200, Australia

(Received December 21, 1995; revised June 6, 1997)

Abstract. In this paper we obtain asymptotic representations of several variance estimators of U-statistics and study their effects for studentizations via Edgeworth expansions. Jackknife, unbiased and Sen’s variance estimators are investigated up to the order o_{p}(n^{−1}). Substituting these estimators to studentized U-statistics, the Edgeworth expansions with remainder term o(n^{−1}) are established and inverting the expansions, the effects on confidence intervals are discussed theoretically. We also show that Hinkley’s corrected jackknife variance estimator is asymptotically equivalent to the unbiased variance estimator up to the order o_{p}(n^{−1}).

Key words and phrases: Edgeworth expansion, Gini’s mean difference, jackknife variance estimator, studentized U-statistics, variance estimation.