Asymptotic Theory of Nonlinear Regression

Let us assume that an observation Xi is a random variable (r.v.) with values in 1 1 (1R1 , 8 ) and distribution Pi (1R1 is the real line, and 8 is the cr-algebra of its Borel subsets). Let us also assume that the unknown distribution Pi belongs to a 1 certain parametric family {{Pi() , () E e}}. We call the triple £i = {{1R1 , 8 , Pi(), () E e}} a statistical experiment generated by the observation Xi. n We shall say that a statistical experiment £n = {{lRn, 8 , P; ,() E e}} is the...
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Let us assume that an observation Xi is a random variable (r.v.) with values in 1 1 (1R1 , 8 ) and distribution Pi (1R1 is the real line, and 8 is the cr-algebra of its Borel subsets). Let us also assume that the unknown distribution Pi belongs to a 1 certain parametric family {{Pi() , () E e}}. We call the triple £i = {{1R1 , 8 , Pi(), () E e}} a statistical experiment generated by the observation Xi. n We shall say that a statistical experiment £n = {{lRn, 8 , P; ,() E e}} is the product of the statistical experiments £i, i = 1, ... ,n if PO' = P () X ... X P () (IRn 1 n n is the n-dimensional Euclidean space, and 8 is the cr-algebra of its Borel subsets). In this manner the experiment £n is generated by n independent observations X = (X1, ... ,Xn). In this book we study the statistical experiments £n generated by observations of the form j = 1, ... ,n. (0.1) Xj = g(j, (}}) + cj, c c In (0.1) g(j, (}}) is a non-random function defined on e , where e is the closure in IRq of the open set e ~ IRq, and C j are independent r. v .-s with common distribution function (dJ.) P not depending on ().