The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
T. W. Anderson
- Vol. 6, Iss: 2, pp 170-176
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The article was published on 1955-02-01 and is currently open access. It has received 552 citations till now. The article focuses on the topics: Convex set & Subderivative.read more
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Adaptive nonparametric drift estimation for diffusion processes using Faber-Schauder expansions
Abstract: We consider the problem of nonparametric estimation of the drift of a continuously observed one-dimensional diffusion with periodic drift. Motivated by computational considerations, van der Meulen e.a. (2014) defined a prior on the drift as a randomly truncated and randomly scaled Faber-Schauder series expansion with Gaussian coefficients. We study the behaviour of the posterior obtained from the prior from a frequentist asymptotic point of view. If the true data generating drift is smooth, it is proved that the posterior is adaptive with posterior contraction rates for the $L_2$-norm that are optimal up to a log factor. Moreover, contraction rates in $L_p$-norms with $p\in (2,\infty]$ are derived as well.
Journal ArticleDOI
On M-estimators and normal quantiles
TL;DR: In this paper, a class of robust estimators of normal quantiles filling the gap between maximum likelihood estimators and empirical quantiles is explored. But their asymptotic variances can be arbitrarily close to variances of the maximum likelihood estimation.
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Finiteness of integrals of functions of Lévy processes
K. Bruce Erickson,Ross Maller +1 more
TL;DR: In this article, necessary and sufficient conditions for the almost sure convergence of the integrals ∫∞ 1 g(a(t)+Mt)df(t), ∫ 1 0 g(mt)+mt)DF(t, and thus of ∫ ∞ 0 g (a(a)) + mt))df (t), where Mt = sup{|Xs| : s ≤ t} is the two-sided maximum process corresponding to a Lévy process (Xt)t≥0, a(·) is a
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Sample Path Properties of Bifractional Brownian Motion
Ciprian A. Tudor,Yimin Xiao +1 more
TL;DR: In this paper, it was shown that the local times of a bifractional Brownian motion are strongly locally non-deterministic and Chung's law of the iterated logarithm for this motion is strongly locally nondeterministic.
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Graph-based regularization for regression problems with alignment and highly-correlated designs.
TL;DR: This work shows how the proposed graph-based regularization yields mean-squared error guarantees for a broad range of covariance graph structures, including block and lattice graphs, which are optimal for many specific covariance graphs.
References
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Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes
TL;DR: In this article, a general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations.
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Theorie der Konvexen Körper
T. Bonnesen,W. Fenchel +1 more
TL;DR: In this article, Minkowski et al. den engen Zusammenhang dieser Begriffbildungen und Satze mit der Frage nach der bestimmung konvexer Flachen durch ihre GAusssche Krtim mung aufgedeckt und tiefliegende diesbeztigliche Satze bewiesen.
Journal ArticleDOI
The Cramer-Smirnov Test in the Parametric Case
TL;DR: In this paper, the authors extended the Cramer-Smirnov and von Mises test to the parametric case, a suggestion of Cramer [1], see also [2].