Mode testing via higher-order density estimation

We investigate the use of fourth-order (reduced bias) variants on kernel density estimation in Silverman’s (J Roy Stat Soc B 43:97–99, 1981) critical bandwidth test of multimodality. Simulations suggest that variable-bandwidth estimates in the style of Abramson (Ann Stat 10:1217–1223, 1982) appear to be especially favorable to mode testing. Conversely, the use of fourth-order kernels leads to extremely poor properties and should be strongly discouraged.     

Comparison of ‘Tail Method’ Exact Confidence Limits for the Difference of Binomial Probabilities

Suppose that Y ₁ and Y ₂ are independent and have Binomial(n ₁,p ₁) and Binomial (n ₂,p ₂) distributions respectively. Also suppose that θ=p ₁−p ₂ is the parameter of interest. We consider the problem of finding an exact confidence limit (either upper or lower) for θ. The solution to this problem is very important for statistical practice in the health and life sciences. The ‘tail method’ provides a solution to this problem. This method finds the exact confidence limit by exact inversion of a hypothesis test based on a specified test statistic. Buehler (J. Am. Stat. Assoc. 52, 482–493, 1957) described, for the first time, a finite-sample optimality property of this confidence limit. Consequently, this confidence limit is sometimes called a Buehler confidence limit. An early tail method confidence limit for θ was described by Santner and Snell (J. Am. Stat. Assoc. 75, 386–394, 1980) who used the maximum likelihood estimator of θ as the test statistic. This confidence limit is known to be very inefficient (see e.g. Cytel Software, StatXact, version 6, vol. 2, 2004). The efficiency of the confidence limit resulting from the tail method depends greatly on the test statistic on which it is based. We use the results of Kabaila (Stat. Probab. Lett. 52, 145–154, 2001) and Kabaila and Lloyd (Aust. New Zealand J. Stat. 46, 463–469, 2004, J. Stat. Plan. Inference 136, 3145–3155, 2006) to provide a detailed explanation for the dependence of this efficiency on the test statistic. We consider test statistics that are estimators, Z-statistics and approximate upper confidence limits. This explanation is used to find the situations in which the tail method exact confidence limits based on test statistics that are estimators or Z-statistics are least efficient.     

Asymptotic normality of the Parzen–Rosenblatt density estimator for strongly mixing random fields

We prove the asymptotic normality of the kernel density estimator (introduced by Rosenblatt, Proc Natl Acad Sci USA 42:43–47, 1956 and Parzen, Ann Math Stat 33:1965–1976, 1962) in the context of stationary strongly mixing random fields. Our approach is based on the Lindeberg’s method rather than on Bernstein’s small-block-large-block technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. Our method allows us to consider only minimal conditions on the bandwidth parameter and provides a simple criterion on the strong mixing coefficients which do not depend on the bandwidth.     

Green function estimate for censored stable processes

 Sharp two-sided estimates for Green functions of censored α-stable process Y in a bounded C ^1,1 open set D are obtained, where α  (1, 2). It is shown that the Martin boundary and minimal Martin boundary of Y can all be identified with the Euclidean boundary of D. Sharp two-sided estimates for the Martin kernel of Y are also derived. Received: 27 January 2002 / Revised version: 10 June 2002 / Published online: 24 October 2002 This research is supported in part by NSF Grant DMS-0071486. Mathematics Subject Classification (2002): Primary: 60J45, 31C35; Secondary: 60G52, 31C15 Keywords or phrases: Censored stable process – Green function – Capacity – Martin boundary – Martin kernel – Harmonic function     

Contributions to the Hypergeometric Function Theory of Heckman and Opdam: Sharp Estimates, Schwartz Space, Heat Kernel

Under the assumption of positive multiplicity, we obtain basic estimates of the hypergeometric functions F _λ and G _λ of Heckman and Opdam, and sharp estimates of the particular functions F ₀ and G ₀. Next we prove the Paley–Wiener theorem for the Schwartz class, solve the heat equation and estimate the heat kernel. Received: June 2006 Revision: December 2006 Accepted: March 2007     

Localization of minimax points

In their proof of Gilbert–Pollak conjecture on Steiner ratio, Du and Hwang (Proceedings 31th FOCS, pp. 76–85 (1990); Algorithmica 7:121–135, 1992) used a result about localization of the minimum points of functions of the type max _y∈Y f(·, y). In this paper, we present a generalization of such a localization in terms of generalized vertices, when we minimize over a compact polyhedron, and Y is a compact set. This is also a strengthening of a result of Du and Pardalos (J. Global Optim. 5:127–129, 1994). We give also a random version of our generalization.     

A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of I s (x)

In high-dimensional directional statistics one of the most basic probability distributions is the von Mises-Fisher (vMF) distribution. Maximum likelihood estimation for the vMF distribution turns out to be surprisingly hard because of a difficult transcendental equation that needs to be solved for computing the concentration parameter κ. This paper is a followup to the recent paper of Tanabe et al. (Comput Stat 22(1):145–157, 2007), who exploited inequalities about Bessel function ratios to obtain an interval in which the parameter estimate for κ should lie; their observation lends theoretical validity to the heuristic approximation of Banerjee et al. (JMLR 6:1345–1382, 2005). Tanabe et al. (Comput Stat 22(1):145–157, 2007) also presented a fixed-point algorithm for computing improved approximations for κ. However, their approximations require (potentially significant) additional computation, and in this short paper we show that given the same amount of computation as their method, one can achieve more accurate approximations using a truncated Newton method. A more interesting contribution of this paper is a simple algorithm for computing I _s (x): the modified Bessel function of the first kind. Surprisingly, our na?ve implementation turns out to be several orders of magnitude faster for large arguments common to high-dimensional data, than the standard implementations in well-established software such as Mathematica ^?, Maple ^?, and Gp/Pari.     

Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization

A fully discrete multi-level spectral Galerkin method in space–time for the two-dimensional nonstationary Navier–Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier–Stokes problem is only solved on the lowest-dimensional space with the largest time step Δt ₁; subsequent approximations are generated on a succession of higher-dimensional spaces with small time step Δt _j by solving a linearized Navier–Stokes problem about the solution on the previous level. Some error estimates are also presented for the J-level spectral Galerkin method. The scaling relations of the dimensional numbers and time mesh widths that lead to optimal accuracy of the approximate solution in H ¹-norm and L ²-norm are investigated, i.e., m _j∼m _j−1 ^3/2 , Δt _j∼Δt _j−1 ^3/2 , j=2,. . .,J. We demonstrate theoretically that a fully discrete J-level spectral Galerkin method is significantly more efficient than the standard one-level spectral Galerkin method. Mathematics subject classifications (2000) 35L70, 65N30, 76D06 Subsidized by the Special Funds for Major State Basic Research Projects G1999032801-07, NSF of China 10371095 and the City University of Hong Kong Research Project 7001093, NSF of China 50323001.     

Faedo–galerkin method for second-order evolution inclusions with <Emphasis Type="Italic">W</Emphasis><Subscript>λ</Subscript>-pseudomonotone mappings

A class of second-order operator differential inclusions with W _λ-pseudomonotone mappings is considered. The problem of the existence of solutions of the Cauchy problem for these inclusions is investigated by using the Faedo–Galerkin method. Important a priori estimates are obtained for solutions and their derivatives. An example that illustrates the proposed approach to the investigation of the problem considered is given.     

On the unsteady Poiseuille flow in a pipe

We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation. G. P. Galdi: Partially supported by the NSF grant DMS–0404834. K. Pileckas: Supported by EC FP6 MCToK program SPADE2, MTKD–CT–2004–014508 A. L. Silvestre: Supported by FCT-Project POCI/MAT/61792/2004     

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality that are independent of Ω. This estimates generalized those of Adimurthi and Yadava (Comm Partial Diff Equ 16(11):1733–1760, 1991) for general p. Here p _* : =  p(N  −  1)/(N  −  p) is the critical exponent for the immersion and N is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of Fernández Bonder and Rossi (Bull Lond Math Soc 37:119–125, 2005). Finally, we study an optimal design problem with critical exponent.      

Estimates for solutions of the nonstationary stokes problem in anisotropic Sobolev spaces with mixed norm

A new method is given to establish L_q,r-estimates for solutions of the nonstationary Stokes problem. The method is based on estimates for heat potentials in these spaces, and it is not connected with investigation of the resolvent for the Stokes operator. It is expected that the method is applicable to a wide class of parabolic initial boundary-value problems. Bibliography:11 titles. Translated fromZapiski Nauchnykh Seminarow POMI, Vol. 222, 1994, pp. 124–151.     

A complete analogue of Hardy’s theorem on SL2(ℝ) and characterization of the heat kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL₂(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝⁿ or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.     

Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel ϕ(t, s) = (t − s)^−μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233: 938–950], the error analysis for this approach is carried out for 0 < μ < 1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-type but also establish the error estimates under a more general regularity assumption on the exact solution.     

Functional a posteriori estimates for Maxwell’s equation

The note is concerned with functional type a posteriori estimates of the difference between exact and approximate solutions of the Maxwell problem . The estimates are derived by transformations of the basic integral identity defining a generalized solution to the problem by using the method suggested by the author. The estimates are obtained in the case ℵ > 0 and ℵ = 0. Bibliography: 10 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 83–88.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

On the asymptotic expansion of the solution of the plane Stokes problem in a perforated domain

This article deals with the asymptotic behavior as ε → 0 of the solution {u _ɛ, p _ɛ} of the plane Stokes problem in a perforated domain. The limit problem is constructed and estimates for the speed of convergence are obtained. It is shown that the speed of convergence is of order O(ε ^3/2). __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 3–20, 2005.     

Remarks on the Cwikel–Lieb–Rozenblum and?Lieb–Thirring Estimates for Schr?dinger Operators on?Riemannian Manifolds

Let M be a general complete Riemannian manifold and consider a Schr?dinger operator −Δ+V on L ²(M). We prove Cwikel–Lieb–Rozenblum as well as Lieb–Thirring type estimates for −Δ+V. These estimates are given in terms of the potential and the heat kernel of the Laplacian on the manifold. Some of our results hold also for Schr?dinger operators with complex-valued potentials.