By Alexander V. Ivanov (auth.)

Let us suppose that an remark Xi is a random variable (r.v.) with values in 1 1 (1R1 , eight ) and distribution Pi (1R1 is the genuine line, and eight is the cr-algebra of its Borel subsets). allow us to additionally think that the unknown distribution Pi belongs to a 1 sure parametric kinfolk {Pi() , () E e}. We name the triple £i = {1R1 , eight , Pi(), () E e} a statistical test generated by way of the remark Xi. n we will say statistical scan £n = {lRn, eight , P; ,() E e} is the manufactured from the statistical experiments £i, i = 1, ... ,n if PO' = P () X ... X P () (IRn 1 n n is the n-dimensional Euclidean area, and eight is the cr-algebra of its Borel subsets). during this demeanour the test £n is generated by means of n self reliant observations X = (X1, ... ,Xn). during this booklet we examine the statistical experiments £n generated by way of observations of the shape j = 1, ... ,n. (0.1) Xj = g(j, (}) + cj, c c In (0.1) g(j, (}) is a non-random functionality outlined on e , the place e is the closure in IRq of the open set e ~ IRq, and C j are autonomous r. v .-s with universal distribution functionality (dJ.) P now not looking on ().

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3): q sup uEs(m) L bi(O + nl/2d;;lu(m)) - bi(O + nl/2d;;lu) dinl i=l 4. DIFFERENTIABILITY OF REGRESSION FUNCTIONS 51 q < (ns*)1/2 g L 'Yi(r*). i=1 Let us denote q 'Y(r*) =L i=1 'Yi(r*). Then p;{ sup II(U) ~ ~} uEQr .. 15), setting is a number which will be chosen later. We need to estimate the probability 1r(~r)' where g = XTn; X = x(r) Consequently > 0, ~r r = 1, ... 18) as being satisfied, for r = 1, ... 19) 52 CHAPTER 1. CONSISTENCY by statement (1) of Theorem AA. 20) where (n = o(n-(S-2)/2) and does not depend upon r.

Then for n > no and C21 = (J-tlja + (1 -,,)~(r) - C20)0I - J-ta > 0, we have PI ~ p;{ n- 1 L lejlOl - J-ta ~ C21} ~ (J-t201 - J-t~)~ln-l by the Chebyshev inequality. 32) let us set R = Ro and " = 2/ PO. P2 < C22 (mod P;). Then by condition IIIq+5 p;{ n- 1 L lejlOl - J-ta ~ C22 } < ( = (J-tlja -2 -1 J-t201 - J-ta2) C22 n , + 2pol ~O)OI - J-ta. And so it remains to estimate the probability (,,' E (0,1)) pnRo ~ In-l/2dn(O)(O~ - 0)1 ~ r} ~ p;{ n- 1 / 0I Ih n (O,u)1 sup _ uEvc(Ro)nu:;(IJ) ~ "'~(r)} + O(n- 1 ).

16) 10 P3 ~ L: p;{ n- Ihn (9, ui)1 ~ (1 - Ph' a(r)} , 1 i=l and consequently it is sufficient to bound each term of the latter summations separately. -s ~jn = IXj - /(j, ui)l. 17) n- 1 L:En~~ (J In n- 1 L:Ene (J In Assuming that JL2 < 00 Dlej < 2s- 1(JLs +n- l 4i sn (Ui,O)) + ,,(8) (ro)) , < 2S - 1(JL8 = JL2 + n- l 4in(ui, 0) < JL2 + x2(ro). v. 21) 3; and its variance + g1)2, 9 ~ 0. Evidently Dlej + gl is a continuous function of g. Let us show that Dlej Since EICj + gl ~ 9-+00 + gl = 9 1 ~ 1'2 9+ -9 P (dx) then Die; + gl + g' -49 / g JL2· + 2 100 xP (dx), g+ (1- (f>(dX»),) + P(dx) [00 xP(dx) -4 ([00 XP(dx))2 -g 19+ < 4 [00 x 2p(dx) ~ 0.