On estimation and detection of a function of infinitely many variables |
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Authors: | Yu I Ingster I A Suslina |
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Institution: | (1) St.Petersburg State Electrotechnical University, St.Petersburg, Russia;(2) St.Petersburg State University of Information Technologies, Mechanics and Optics, St.-Petersburg, Russia |
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Abstract: | We observe an unknown function of infinitely many variables f = f(t), t = (t1, ..., tn, ... ) ∈, 0, 1]∞, in the Gaussian white noise of level ε > 0. We suppose that in each variable there exists a 1-periodical σ-smooth extension
of the function f(t) to IR
∞. Taking a quantity σ > 0 and a positive sequence a = {ak}, we consider the set
that consists of functions f such that
. We consider the cases ak = kα and ak = exp(λk), α > 0, λ > 0. We would like to estimate a function f ∈
or to test the null hypothesis H0: f = 0 against the alternatives f ∈
, where the set
consists of functions f ∈
such that ∥f∥2 ≥ r. In the estimation problem, we obtain the asymptotics (as ε → 0) of the minimax quadratic risk. In the detection problem,
we study the sharp asymptotics of minimax separation rates f
ɛ
*
that provide distiguishability in the problems. Bibliography: 12 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 91–113. |
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Keywords: | |
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