By Alexander V. Ivanov (auth.)

Let us imagine 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 imagine that the unknown distribution Pi belongs to a 1 yes parametric relations {Pi() , () E e}. We name the triple £i = {1R1 , eight , Pi(), () E e} a statistical scan generated through the statement 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 scan £n is generated via n self sufficient observations X = (X1, ... ,Xn). during this booklet we learn 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 sustaining r. v .-s with universal distribution functionality (dJ.) P now not looking on ().

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**Sample text**

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. ll + Po"'an - l CPn (8 + u, 8). 14). e. e .. The estimator lo: of the parameter 0 E e, obtained from the observations Xl, ... e .. 29) ,u2EvC(R)nU~ (0) IU 1-u21

The relation between the smoothness of the function g(j,O) and the condition IlIq+6 is somewhat more complicated. Let us set o E e. The symmetric matrix 1(0) is non-negative definite. Let Amin(I(O)) be the smallest eigenvalue of the matrix 1(0). We now introduce a condition which plays an important role in the following chapters: v. 7) If a second derivative exists for the function g(j, 0), then let us set diln(O) = ( Lg~I(j,O) ) 1/2 , i,l = 1, ... ,q. II6 • The functions gi(j,O), i = 1, ... 8) = 1, ...

CONSISTENCY 58 or 5 STRONG CONSISTENCY Let us consider the statistical experiment £ N = {R N , EN, pf, () E e} generated by the sequence of observations X 1 ,X2, ... of the form (0,1). ) convergence of the statistical estimators to a true value of the parameter () E e. It is understood that all assertions of the preceding sections can be reformulated for the experiments considered. Let On = On(X1" " ,Xn), n ~ 1, be a certain sequence of estimators. ) . -s On. Let us mention two examples. Let us assume that in the conditions of Theorem 8 {Ls < 00 for some s > 4.