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X m, = ,y" .. ,y m2 I ) X (n,) ,Y (n2) E(E(h(X" ... , XIn, ,y" ... ,y m )IFnl-'n 2 2 E (h ( X" U 38 ••• , X rvn"svn2 In, (h), , Y, , ••• , Y In 2 (n 2) - measurable and if h ( X . , ••• , X . =1 I In I ) Fn v , Ir r, s) r n vs) , 2 I ) Fr s) ' A ». since f Now let r,s h n f n 1,n 2 be degenerate. •• , u, ... ) dF (u ) then E(h( . X. letry . ': Let h be adegenerate generalizec k e r n e l of degree (rn"m ) . 2 If (1. 4. 5) IIhll 2 := E h 2 (X" •.. , x m, ,y" ... ,y then Var U n"n 2 m2 ) < 00, (h) Proof: Since Eh( X X· , y" 1,···, m, Var U n n (h ) 1,n 2 n (m~)-2(m~)-2 ...

We shall now discuss a few examples in the remaining pa rt of this section. Some basic facts from calculus are not explicit- ly stated but are obvious in each case . For example, condi- tions for interchanging differentiation and integration, the fact that S f' (x) dx 0 for a differentiable density f etc. We shall not give all computations in each example. 1: Let Sx T(F) = where the integral exists. dF(x) be defined for all F T is called the empirical expec- This set clearly is star shaped and tation.

Mc } f or some l :;; i ~ c, of { I , .. , m}, d efine g r (x 1 , · · · , x c ) = h (x 1 , · .. , x 1, x 2'· .. , x c) where xi (1 S i S c) app ears m times i ment in h. 30 ~uccessively as an argu- Then the seeond sum equals n- m L I m! (n) U ( ) m ! m ! e n g1 . e 1 Eaeh g1 is symmetrie. g1(X 1"",Xe) = To see this fix indices i and j. Then h(X1""'Xi-1'~i"",xi'Xi+1' ... m. l Xj_1~Xj"",Xj"",xe) m. x e)· • '---v------' m. m. J l The arguments in both right-hand sides are apermutation of eaeh other, and sinee h is symmetrie the right-hand sides eoineide.

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