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Том 25, № 4 (2016)
- Год: 2016
- Статей: 5
- URL: https://journal-vniispk.ru/1066-5307/issue/view/13890
Article
Optimal methods of interpolation in Nonparametric Regression
Аннотация
Within the framework of Optimal Recovery, optimal methods of interpolation, based on the Abel–Jacobi elliptic functions, have been found for some Hardy classes of analytic functions [9]. It will be shown that these methods are also optimal according to criteria of Optimal Design and Nonparametric Regression.
For all noise levels away from 0, the mean squared error of the optimal interpolant is evaluated explicitly, in a non-asymptotic setting. In this result, a pivotal role is played by an interference effect in which both stochastic and deterministic parts of the interpolant exhibit an oscillating behavior, with the two oscillating functions canceling each other.
235-261
Rate of convergence of truncated stochastic approximation procedures with moving bounds
Аннотация
The paper is concerned with stochastic approximation procedures having three main characteristics: truncations with random moving bounds, a matrix-valued random step-size sequence, and a dynamically changing random regression function. We study convergence and rate of convergence. Main results are supplemented with corollaries to establish various sets of sufficient conditions, with the main emphasis on the parametric statistical estimation. The theory is illustrated by examples and special cases.
262-280
Some applications of the strong approximation of the integrated empirical copula processes
Аннотация
The purpose of the present paper is to provide a strong invariance principle for the integrated empirical copula process [introduced in a series of papers by Henze and Nikitin in the univariate setting] with the rate of the approximation for multivariate empirical processes. The applications discussed here are change-point detection in multivariate copula models and the integrated empirical copula process with estimated parameter. Finally, a general notion of bootstrapped integrated empirical copula process, constructed by exchangeably weighting sample, is presented.
281-303
The minimum increment of f-divergences given total variation distances
Аннотация
Let (Pi, Qi), i = 0, 1, be two pairs of probability measures defined on measurable spaces (Ωi,Fi) respectively. Assume that the pair (P1, Q1) is more informative than (P0,Q0) for testing problems. This amounts to say that If (P1,Q1) ≥ If (P0,Q0), where If (·, ·) is an arbitrary f-divergence. We find a precise lower bound for the increment of f-divergences If(P1,Q1) − If(P0,Q0) provided that the total variation distances ||Q1 − P1|| and ||Q0 − P0|| are given. This optimization problem can be reduced to the case where P1 and Q1 are defined on the space consisting of four points, and P0 and Q0 are obtained from P1 and Q1 respectively by merging two of these four points. The result includes the well-known lower and upper bounds for If(P,Q) given ||Q − P||.
304-312
Cramér type moderate deviations for trimmed L-statistics
Аннотация
We establish Cramér type moderate deviation results for heavy trimmed L-statistics; we obtain our results under a very mild smoothness condition on the inversion F−1 (F is the underlying distribution function of i.i.d. observations) near two points, where trimming occurs, we assume also some smoothness of weights of the L-statistic. Our results complement previous work on Cramér type large deviations for trimmed L-statistics [8] and [5].
313-322