Adaptive Optimal Nonparametric Regression and Density Estimation Based on Fourier-Legendre Expansion

Ostrovsky, E.; Zelikov, Y.

doi:10.48550/arxiv.0706.0881

Cited by 3 publications

(9 citation statements)

References 24 publications

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“…The proposition (6.6) is proved in the article [11]; see also [9]. We will formulate the main result of this section, which may be obtained after simple calculations basing on the lemma 6.1.…”

Section: T) Then We Have For the Functionmentioning

confidence: 87%

“…Recently, see [2], [3], [4], [5], [6], [7], [9], [10], [11], etc. appears the so-called Grand Lebesgue Spaces The set of all ψ functions with support supp(ψ) = (A, B) will be denoted by Ψ(A, B).…”

Section: T) Then We Have For the Functionmentioning

confidence: 99%

“…appears the so-called Grand Lebesgue Spaces The set of all ψ functions with support supp(ψ) = (A, B) will be denoted by Ψ(A, B). This spaces are rearrangement invariant, see [1], and are used, for example, in the theory of probability [7], [10], [11]; theory of Partial Differential Equations [3], [6]; functional analysis [4], [5], [9], [11]; theory of Fourier series [10], theory of martingales [11],mathematical statistics [22], [23]; theory of approximation [15] etc.…”

Section: T) Then We Have For the Functionmentioning

confidence: 99%

“…A possible applications of tail estimates: Functional Analysis, see the classical books of C.Bennet and R.Sharpley [1], S.G. Krein Yu.V. Petunin and E.M. Semenov [8], and also in the articles [16], [17], [18]; Probability Theory [19], [21]; Numerical Methods Monte-Carlo [20]; Statistics [10], [11], [12], theory of random processes and fields [7], [13] etc.…”

mentioning

confidence: 99%

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Tchebyshev's characteristic of rearrangement invariant space

Ostrovsky,

Sirota

2012

Preprint

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We introduce and investigate in this short article a new characteristic of rearrangement invariant (r.i.) (symmetric) space, namely so-called Tchebychev's characteristic.We reveal an important class of the r.i. spaces -so called regular r. i. spaces and show that the majority of known r.i. spaces: Lebesgue-Riesz, Grand Lebesgue Spaces, Orlicz, Lorentz and Marcinkiewicz r.i. spaces are regular. But we construct after several examples of r.i. spaces without the regular property.Applications -Probability theory and Statistics.

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Section: T) Then We Have For the Functionmentioning

confidence: 87%

Section: T) Then We Have For the Functionmentioning

confidence: 99%

Section: T) Then We Have For the Functionmentioning

confidence: 99%

mentioning

confidence: 99%

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Tchebyshev's characteristic of rearrangement invariant space

Ostrovsky,

Sirota

2012

Preprint

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“…It is interest in our opinion to deduce the recursive projection density (and regression, spectral density) function estimates for the so -called adaptive estimates, having however the optimal rate of convergence, which does not dependent on the unknown, in general case, class of smoothness for estimating function. See for example [4], [21], [22].…”

Section: Assume In Addition Thatmentioning

confidence: 99%

Exponential confidence region based on the projection density estimate. Recursivity of these estimations

Formica¹,

Ostrovsky²

2021

Preprint

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We investigate the famous Tchentzov's projection density statistical estimation in order to deduce the exponential decreasing tail of distribution for the natural normalized deviation.We modify these estimations assuming the square integrability of estimated function, to make it recursive form, which is more convenient for applications, however they have at the same speed of convergence as the for the classical ones in the composite Hilbert space norm.

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Optimal Adaptive Nonparametric Denoising of Multidimensional - Time Signal

Ostrovsky,

Sirota

2008

Preprint

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We construct an adaptive asymptotically optimal in the classical norm of the space L(2) of square integrable functions non -parametrical multidimensional time defined signal regaining (adaptive filtration, noise canceller) on the background noise via multidimensional truncated Legendre expansion and optimal experience design.The two -dimensional case is known as a picture processing, picture analysis or image processing.We offer a two version of an confidence region building, also adaptive. Our estimates proposed by us have successfully passed experimental tests on problem by simulate of modeled with the use of pseudo-random numbers as well as on real data (of seismic signals etc.) for which our estimations of the different signals were compared with classical estimates obtained by the kernel or wavelets estimations method. The precision of proposed here estimations is better.Our adaptive truncation may be used also for the signal and image compression.

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Adaptive Optimal Nonparametric Regression and Density Estimation Based on Fourier-Legendre Expansion

Cited by 3 publications

References 24 publications

Tchebyshev's characteristic of rearrangement invariant space

Tchebyshev's characteristic of rearrangement invariant space

Exponential confidence region based on the projection density estimate. Recursivity of these estimations

Optimal Adaptive Nonparametric Denoising of Multidimensional - Time Signal

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