Best Possible Rate Of Convergence Research Articles

Discrimination distance bounds and statistical applications

Bounds are obtained for the Kullback-Leibler discrimination distance between two random vectorsX andY. IfX is a sequence of independent random variables whose densities have similar tail behavior andY=AX, whereA is an invertible matrix, then the bounds are a product of terms depending onA andX separately. We apply these bounds to obtain the best possible rate of convergence for any estimator of the parameters of an autoregressive process with innovations in the domain of attraction of a stable law. We provide a general theorem establishing the link between total variation proximity of measures and the rate of convergence of statistical estimates to complete the exposition for this application.

On Optimal Algorithms in an Asymptotic Model with Gaussian Measure

We study the approximate solution of linear problems in separable Hilbert spaces equipped with a Gaussian measure. We find information and algorithms with the best possible rate of convergence. Although adaptive information and nonlinear algorithms are permitted, we prove that nonadaptive information and linear algorithms are optimal. An algorithm is optimal if it converges with a rate of convergence that is no worse than the rate of any other algorithm except on sets of measure zero. We prove that algorithms and information that minimize the average errors lead to the best possible rate of convergence. This exhibits a close relation between the asymptotic and average case models.

On one sequence of iteration parameters

An infinite sequence of iteration parameters is constructed for which the corresponding iterative process for the solution of a linear operator equation in Hilbert space has a rate of convergence close to the best possible rate of convergence.