Empirical Estimates in Stochastic Optimization and Identification

Some applied problems from random field theory

“…As the criterion functions characterizing the estimation quality, maximum likelihood, least squares, and least absolute values are usually taken. Monographs [2,4,5,10] describe these problems in details. Moreover, they contain the results on the asymptotic behavior of the estimates obtained (consistency condition, asymptotic distribution, weak convergence of the measures generated by these estimates).…”

“…If we denote a a n n J = arg min ( ) , a a * ( ) = arg min J , then the task is to find the conditions for a a n ® *, J J n n ( ) ( *) a a ® in some probabilistic sense. Such studies were carried out in [4][5][6][7][8][9][10] under rather general conditions for the criterion function, function (2) being its special case.…”

“…For example, the radiation of a heated body (gas) is the total radiation of atoms acting as oscillators, their number is of order 10 20 . Therefore, the need arises to analyze a periodic signal x t ( ) represented as an absolutely converging series If the signal x t ( )is transmitted along a communication channel with the additive noise n t ( ), and the parameters A and w are unknown, we arrive at the identification model (estimating the unknown parameters A and w) studied in [1, 4,10]. The model with a homogeneous random field [4] is considered similarly.…”

Some approaches to financial risk assessment

“…Note that it has become widely popular in recent years and many publications appeared (see [2][3][4]). …”

“…Dorogovtsev [5] presents a somewhat different formulation of Theorem 2. Theorems 1 and 2 allow deriving the following statements proposed in [2][3][4] and proved in somewhat different formulations in [6][7][8][9]. (ii) for any c > 0 , E fx…”