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Vol 26, No 1 (2017)
- Year: 2017
- Articles: 5
- URL: https://journal-vniispk.ru/1066-5307/issue/view/13892
Article
On the lower bound in second order estimation for Poisson processes: Asymptotic efficiency
Abstract
In the estimation problem of the mean function of an inhomogeneous Poisson process there is a class of kernel type estimators that are asymptotically efficient alongside with the empirical mean function. We start by describing such a class of estimators which we call first order efficient estimators. To choose the best one among them we prove a lower bound that compares the second order term of the mean integrated square error of all estimators. The proof is carried out under the assumption on the mean function Λ(·) that Λ(τ) = S, where S is a known positive number. In the end, we discuss the possibility of the construction of an estimator which attains this lower bound, thus, is asymptotically second order efficient.
1-19
Saddlepoint approximations to the distribution of the total distance of the multivariate isotropic and von Mises–Fisher random walks
Abstract
This article considers the random walk over Rp, with p ≥ 2, where the directions taken by the individual steps follow either the isotropic or the vonMises–Fisher distributions. Saddlepoint approximations to the density and to upper tail probabilities of the total distance covered by the random walk, i.e., of the length of the resultant, are derived. The saddlepoint approximations are onedimensional and simple to compute, even though the initial problem is p-dimensional. Numerical illustrations of the high accuracy of the proposed approximations are provided.
20-36
Asymptotic behavior of truncated stochastic approximation procedures
Abstract
We study asymptotic behavior of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix-valued random step-size sequence, and a dynamically changing random regression function. In particular, we show that under quitemild conditions, stochastic approximation procedures are asymptotically linear in the statistical sense, that is, they can be represented as weighted sums of random variables. Therefore, a suitable formof the central limit theoremcan be applied to derive asymptotic distribution of the corresponding processes. The theory is illustrated by various examples and special cases.
37-54
An oracle inequality for quasi-Bayesian nonnegative matrix factorization
Abstract
The aim of this paper is to provide some theoretical understanding of quasi-Bayesian aggregation methods of nonnegative matrix factorization. We derive an oracle inequality for an aggregated estimator. This result holds for a very general class of prior distributions and shows how the prior affects the rate of convergence.
55-67
A supermartingale argument for characterizing the functional Hill process weak law for small parameters
Abstract
The paper deals with the asymptotic laws of functionals of standard exponential random variables. These classes of statistics are closely related to estimators of the extreme value index when the underlying distribution function is in theWeibull domain of attraction.We use techniques based on martingales theory to describe the non-Gaussian asymptotic distribution of the aforementioned statistics.We provide results of a simulation study as well as statistical tests that may be of interest with the proposed results.
68-80