Download e-book for iPad: Asymptotic Theory of Nonlinear Regression by Alexander V. Ivanov (auth.)

By Alexander V. Ivanov (auth.)

ISBN-10: 9048147751

ISBN-13: 9789048147755

ISBN-10: 9401588775

ISBN-13: 9789401588775

Let us imagine that an statement Xi is a random variable (r.v.) with values in 1 1 (1R1 , eight ) and distribution Pi (1R1 is the true line, and eight is the cr-algebra of its Borel subsets). allow us to additionally imagine that the unknown distribution Pi belongs to a 1 yes parametric family members {Pi() , () E e}. We name the triple £i = {1R1 , eight , Pi(), () E e} a statistical scan generated via the statement Xi. n we will say statistical scan £n = {lRn, eight , P; ,() E e} is the made of 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 through 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 self sufficient r. v .-s with universal distribution functionality (dJ.) P no longer counting on ().

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Let us write 4»kn(Ul, U2) 4»On(Ul,U2) And so 4»2n I~. tB IIIq+4. = 4». J( l~J~n k = 1,2, ... , IfU,ud - f(j,U2)1, Let us assume that: < 00 for some natural number s. 6. 0 > 0 are some numbers. lIs. (1) For any C > 0 and r > 0 there exists 8 = 8(r,c) such that sup sup n- 14»1n(U1,U2) ::; c. 17) (JET uEvC(r)nu:;«(J) sup sup 4»on{u,O) ::; x(1), • s = 1. v .. If the assumptions I~, lIs, IIIq+4 are satisfied, then for any r > 0 THEOREM sup P;{ldn {()) {On - ())I ~ rn 1 / 2 } (JET ={ O{n-B+1) 0(1), ' s>2 8 -, = 1.

9) if H ~ Zn. ~O(Zn)H-l+l < Cl6'li;;2s(H)HO:s 10 < pO:S dp c17H-lw;;2s(H)w~'YS(zn) < C17 H - lW;;2S(l-'Y)(H). 1: In the proof of Theorem 7 the relation (3) of condition IIIq+2 is not used directly. It shows, however, that we are not justified in arguing also as in the proof of Theorem 6. In fact, if (3) is satisfied then for x > Zn and n > no XO:W;;2(X) If, for example, ~ cg2 XO:Z;;2/3. x n -- z2/3/O: n then x~w;;2(xn) does not tend to zero as n REMARK > Z n, -t 00. 2: Let us assume that E> is a bounded set and where d(E» = sup z,yE9 Ix - yl, • 30 CHAPTER 1.

In this case, for any C > 0 supPJ'{n- 1 Ihn (O,Ui)1 ~ c} (JET --t n-too 0, • which completes the proof of the Theorem. L1 + 2X(1) (ro) (modPJ'). L1 - 2X(1) (ro)} J1z1>gln IxlP (dx) --t n-too 0, CHAPTER 1. CONSISTENCY 38 where c:' E (0, c:) is some number. 6 is satisfied. Let us next verify that condition (2) is satisfied for T = 1: < E;[c:~ + 21C:i Ix(1) (ro) + (x(1) (ro))2 + (1-£1 + x(1) (ro))2] xX{IC:il < n < ( + 1-£1 + 2X(1) (ro)} . &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.

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Asymptotic Theory of Nonlinear Regression by Alexander V. Ivanov (auth.)

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