By Alexander V. Ivanov (auth.)

Let us suppose that an statement 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 suppose that the unknown distribution Pi belongs to a 1 yes parametric kinfolk {Pi() , () E e}. We name the triple £i = {1R1 , eight , Pi(), () E e} a statistical test generated via the statement Xi. n we will say statistical test £n = {lRn, eight , P; ,() E e} is the made from the statistical experiments £i, i = 1, ... ,n if PO' = P () X ... X P () (IRn 1 n n is the n-dimensional Euclidean house, and eight is the cr-algebra of its Borel subsets). during this demeanour the scan £n is generated by means of n self sustaining observations X = (X1, ... ,Xn). during this ebook we research the statistical experiments £n generated by means 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 self reliant r. v .-s with universal distribution functionality (dJ.) P now not looking on ().

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1 +2x(1) (ro) x2P(dx)+41-£1x(1)(ro)+2(x(1)(ro))2+1-£~. &1+2x(1)(ro) x 2 P (dx) ----t n-too O. n IxlP (dx) ----t n-too O. = 1) let us write for 9 E T By analogy with Theorem 8 we can make the following remark. 1: Let dn (9) = n 1 / 2 1q, e and be a closed set. Then Theorem 9 can be formulated with a simpler discrimination condition for the parameters than III q+4. 6. (r). 26) 3. 26) are awkward to verify. 26) to be fulfilled, for example. Let us assume that the dJ. v. Cj has a Lebesgue decomposition "'a> with absolutely continuous components Pa, (1) sup (2) inf Pa(X) 1"'1::;90 sup j2::l 01002E9C P~ = Ig(j,8l ) - 0 Pa and g(j,82 )1 = go < 00, = Po > O.

J cos J 2+4 sin(J2 ' and uniformly with respect to (J E T. On the other hand, d2n «(J) = (J1 (Lj2 sin2 (J2j y/2 (J1 (n3 6 + n 2 _ ~ + n 2 sin(2n + 1)(J2 + ~ 4 12 4 sin (J2 cos2n(J2 4 sin2 (J2 _! j6 (1 + O(n-1)) n uniformly with (J E T. Therefore it is appropriate to take the matrix 4. DIFFERENTIABILITY OF REGRESSION FUNCTIONS 55 as the normalising matrix instead of dnO. 2 + ~4n (sin(2~sm0+2 1)02 _ 1)) 01(0 1 + v'2 u 1) 2n sin(n + X ( ~) (20 + oV:n u 2 2 ) ------;-"------,--~ (2 v'6- u sin 0 + 20 1 n 2) + • (n sm + 1) v'6 2 2 0 1 U .

CONSISTENCY by statement (1) of Theorem AA. 20) where (n = o(n-(S-2)/2) and does not depend upon r. 15): P1 = < P; {t Ibi(O + n1/2d;;-lu(m) Idi; :2 t5xox r: 1 n 1/ 2Tn } L P;{bi(O + n 1/ 2d;;-l q U (m)Idfu1(0 + n1/2d;;-lu(m) :2 p,~/2ain} , i=l where ~ 2( P,11/2(r ain = uXOXT + 1) q(3-i (*))-11 r og1/2 n, i = 1, ... ,q. 5. 5, c. -2 1/-1/2 cJ'f t'(J' , u(m»)n1/2d-1(O+n1/2d-1(O)u(m») <,,In in n , j = 1, ... 2). Therefore 4. DIFFERENTIABILITY OF REGRESSION FUNCTIONS 53 where the constants xi(T) do not depend upon r.