Optimal convergence properties of kernel estimates of derivatives of a density function

1979 ◽  
pp. 144-154 ◽  
Author(s):  
Hans-Georg Müller ◽  
Theo Gasser
Keyword(s):  
Density Function ◽  
Convergence Properties ◽  
Kernel Estimates ◽  
Optimal Convergence ◽  
Derivatives Of

scholarly journals Estimation of Partial derivatives of the average of densities belonging to a family of densities

1975 ◽  
Vol 20 (2) ◽  
pp. 230-241 ◽  
Author(s):  
V. Susarla ◽  
S. Kumar
Keyword(s):  
Lebesgue Measure ◽  
Empirical Bayes ◽  
Mean Squared Error ◽  
Asymptotic Properties ◽  
Convergence Properties ◽  
Kernel Estimates ◽  
Partial Derivatives ◽  
Squared Error ◽  
The Mean ◽  
Derivatives Of

Recently, attention has been drawn to the problem of estimation of a k-variate probability density and its partial derivatives of various orders. Specifically, let X1, …, Xn be i.i.d. k-variate random variables with common density f wrt Lebesgue measure μ on the k-dimensional σ-field Bk. Parzen (1962) in the k = 1 case and Cacoullos (1966) in the k ≧ 1 case gave the asymptotic properties of a class of kernel estimates fn(x), x ∈ Rk, of f(x) based on X1, …, Xn. The asymptotic properties given in the above two papers concern consistency, asymp-totic unbiasedness, bounds for the mean squared error and asymptotic normality of fn. Also in the context of an empirical Bayes two-action problem, Johns and Van Ryzin (1972) introduced kernel estimates for f(x) and the derivative f'(x)for x∈R1 when f is a mixture of univariate exponential densities wrt Lebesgue measure on B1. They also investigated the asymptotic unbiasedness and themean squared error convergence properties of these estimates. Lin (1968) statedsome generalizations of the results of Johns and Van Ryzin, with applicationsto empirical Bayes decision problems.

scholarly journals On the integral kernels of derivatives of the Ornstein–Uhlenbeck semigroup

2016 ◽  
Vol 19 (04) ◽  
pp. 1650030 ◽  
Author(s):  
Jonas Teuwen
Keyword(s):  
Closed Form ◽  
Hermite Polynomials ◽  
Closed Form Expression ◽  
Alternative Proof ◽  
Kernel Estimates ◽  
Form Expression ◽  
Mehler Kernel ◽  
Derivatives Of

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup [Formula: see text]. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers [Formula: see text] of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for [Formula: see text]. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.

scholarly journals Harmonic Besov spaces on the ball

2016 ◽  
Vol 27 (09) ◽  
pp. 1650070 ◽  
Author(s):  
Seçil Gergün ◽  
H. Turgay Kaptanoğlu ◽  
A. Ersin Üreyen
Keyword(s):  
Besov Spaces ◽  
Bergman Spaces ◽  
Reproducing Kernels ◽  
Integral Representations ◽  
Kernel Estimates ◽  
Harmonic Bergman Spaces ◽  
Infinite Sums ◽  
Bergman Projections ◽  
Derivatives Of

We initiate a detailed study of two-parameter Besov spaces on the unit ball of [Formula: see text] consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem.

HEDGING STRATEGY WITH LANGEVIN EVOLUTION

2000 ◽  
Vol 03 (03) ◽  
pp. 553-556
Author(s):  
S. MARIANI ◽  
G. ROTUNDO ◽  
B. TIROZZI
Keyword(s):  
Density Function ◽  
Langevin Equation ◽  
Additive Noise ◽  
Market Model ◽  
Infinite Variance ◽  
Kernel Estimates ◽  
Hedging Strategy ◽  
Market Data ◽  
Price Fluctuations ◽  
Levy Distribution

In recent years there has been much attention paid to pricing and hedging models that are more general than the Black and Scholes' one, whose hypotheses often aren't satisfied by true market data. Sato and Takayasu proposed a market model that can produce price fluctuations with infinite variance from a deterministic behaviour of many market's dealers as it emerged from their simulations. We check the possibility to apply this model to an Italian stock, the "Assicurazioni Generali" and determine the density function of the multiplicative and additive noise appearing in the proposed Langevin equation. We find also the Lévy distribution for the changes. This distribution is consistent with the histograms and the Kernel estimates. These results are used for hedging following the approach of Sornette and Bouchaud.

scholarly journals Density Function of the FK4 Systematic Errors

1990 ◽  
Vol 141 ◽  
pp. 94-94
Author(s):  
V. S. Gubanov ◽  
N. I. Solina
Keyword(s):  
Density Function ◽  
Spherical Harmonics ◽  
Unit Sphere ◽  
Systematic Errors ◽  
Layer Potential ◽  
Partial Derivatives ◽  
Derivatives Of

A notion of density function of systematic errors of astrometric catalogues distributed on the unit sphere as a simple layer is introduced. Components of the catalogue errors in any direction are determined as partial derivatives of layer potential in the same direction. For example, the density function of FK4 errors is computed as an expansion of spherical harmonics.

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