The rate of the normal approximation for jackknifingU-statistics

LetU _n be aU-statistic with symmetric kernelh(x, y) such thatEh(X₁, X₂)=θ and VarEh(X₁, X₂)?θ|X ₁]>0. Letf(x) be a function defined onR andf″ be bounded. Iff(θ) is the parameter of interest, a natural estimator isf(U_n). It is known that the distribution function of \(z_n = \frac{{\sqrt n \{ Jf(Un) - f(\theta )\} }}{{S_n^* }}\) converges to the standard normal distribution Φ(x) asn→∞, whereJf(U_n) is: the jackknife estimator off(U_n), andS _n ^*2 is the jackknife estimator of the asymptotic variance ofn ^1/2 Jf(U_n). It is of theoretical value to study the rate of the normal approximation of the statisticz _n. In this paper, assuming the existence of fourth moment ofh(X₁, X₂), we show that $$\mathop {\sup }\limits_x |P\{ z_n \le x\} - \Phi (x)| = O(n^{ - 1/2} \log n).$$ This improves the earlier results of Cheng (1981).