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

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 definite parametric relations {Pi() , () E e}. We name the triple £i = {1R1 , eight , Pi(), () E e} a statistical test generated by way of the statement Xi. n we will say statistical test £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 test £n is generated via n self sufficient observations X = (X1, ... ,Xn). during this booklet we examine 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 sufficient r. v .-s with universal distribution functionality (dJ.) P no longer counting on ().

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S On. Let us mention two examples. Let us assume that in the conditions of Theorem 8 {Ls < 00 for some s > 4. Then for any r > 0 and () E T n and, consequently, Analogously, under the conditions of Theorem 9, if number s ~ 3, then for any () E T {Ls < 00 for some natural In fact, these estimators are strongly consistent for less severe constraints. ) for non-identically distributed observations. 59 5. STRONG CONSISTENCY Let us consider a sequence of families of Borel functions Fj = {fj( . ,0) :]R1 Clearly, for any 01 , O2 E -t iiP, 0 E eel, e and j .

III q+6. 4) "0. Let us denote fi (j, U) = gi(j, gh + n 1/ 2d;;1 (O)u), cp~)(Bl,B2) = 2:[gi(j,Bt} - gi(j,02)J2, 4i~)(Ul,U2) = 2:[Ii(j,ut} - IV t . ; (B) sup_ (JET uEv(R)nU,,((J) Ul,U2 E U~(B), fi(j,u2)J2, 3 and any R m B1,B2 E e, i = 1, ... ,q. >0 L 11i(j, u)l t < 00, i = 1, ... , q. 1) there follows a condition that makes the requirement II2 of Theorem 8 more precise. 6) ,u2Eve(R)nu~ ((J) where 1,8(R)1 is the norm of the vector ,8(R) = (i31(R), ... ,i3q (R)). Proof: Let BET be fixed. By the finite increments formula for Ul, U2 E vC(R) n U:;(B), with the aid of the Cauchy-Bunyakovski inequality we find n- 14iN(ul,U2) = 2n- 1 L(j(j, Ul) - f(j, Ul x (\1 f(j, Ul + TJn (U2 + TJn(U2 - Ul))) - ut}), nl/2d;;1 (B) (U2 - Ul)} 4.

S On. Let us mention two examples. Let us assume that in the conditions of Theorem 8 {Ls < 00 for some s > 4. Then for any r > 0 and () E T n and, consequently, Analogously, under the conditions of Theorem 9, if number s ~ 3, then for any () E T {Ls < 00 for some natural In fact, these estimators are strongly consistent for less severe constraints. ) for non-identically distributed observations. 59 5. STRONG CONSISTENCY Let us consider a sequence of families of Borel functions Fj = {fj( . ,0) :]R1 Clearly, for any 01 , O2 E -t iiP, 0 E eel, e and j .

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