The Asymptotic Normality of Linear Functions of Conditional Order Statistics

<正> IntroductionLet (X, Y) be a Rd x R1 -valued random vector and F(y \ x) be the conditional distribution of Y given X = x. In recent years many statisticians make systematic study to the non-parametricestimation of the regression functions m(x) = E(Y\ X — x) = y.F(dy|#). For example Stone'1',Devroye and Wagner, and Devroye, etc. However, it is well known that the conditional expectation is not robust. But if the conditional distribution is symmetric about m(x), i.e. there exists a symmetrical distribution function H(-) such that F(y\x) = H{y — rn(x)), then m(x) turns