On some asymptotic properties of U statistics and one-sided estimates

Abstract

Let {X_i,1≤i≤n} be independent and identically distributed random variables. For a symmetric function h of m arguments, with θ=Eh(X₁,…X_m), we propose estimators θ_nof θ that have the property that θ_n→θ almost surely (a.s.) and θ_n≥θ a.s. for all large n. This extends the results of Gilat and Hill, who proved this result for θ=Eh(X₁). The proofs here are based on an almost sure representation that we establish for U statistics. As a consequence of this representation, we obtain the Marcinkiewicz Zygmund strong law of large numbers for U statistics and for a special class of L statistics.