Nonnegative Kernel Research Articles

A Variation of the Jentzsch Theorem for a Symmetric Integral Kernel and its Application.

A variation of the Jentzsch theorem is given; the largest characteristic value with respect to odd characteristic functions is simple for an integral equation whose kernel is non-negative, symmetric, even on one of variables in R n and having some other conditions. In order to prove the new theorem, we present a simple proof for the Jentzsch theorem for a non-negative and symmetric integral kernel. We give an application of the new theorem to correlation functions of the one-dimensional spin systems.

Classes of pairs of meromorphic matrix valued functions generated by nonnegative kernels and associated Nevanlinna-Pick problems

Families of pairs of matrix-valued meromorphic functions $$\mathcal{P}$$ (ρ,P) depending on two parameters ϱ andP are introduced. They are the projective analogues of classes of functions studied in [1] and include as special cases the projective Schur, Nevanlinna and Caratheodory classes. A two sided Nevanlinna-Pick interpolation problem is defined and solved in $$\mathcal{P}$$ (ρ,P), using the fundamental matrix inequality method.

A Geometric Combination Estimator for $d$-Dimensional Ordinal Sparse Contingency Tables

A geometric combination estimator is proposed for $d$-dimensional ordinal contingency tables. The proposed estimator is nonnegative. It is shown that, assuming sufficient smoothness and boundary conditions for the underlying probabilities, the rate of convergence of mean summed squared error (MSSE) of this estimator is $O(K^{-1}N^{-8/(d+8)})$ for $d$-dimensional tables $(d \leq 4)$ with $K$ cells and sample size $N$. This rate is optimal under the smoothness assumptions, and is faster than that attained by nonnegative kernel estimates. Boundary kernels for multidimensional tables are also developed for the proposed estimator to relax restrictive boundary conditions, resulting in summed squared error (SSE) being of order $O_p(K^{-1}N^{-8/(d+8)})$ for all $d \geq 1$. The behavior of the new estimator is investigated through simulations and applications to real data. It is shown that even for relatively small tables, these estimators are superior to nonnegative kernel estimators, in sharp contrast to the relatively unimpressive performance of such estimators for continuous data.

Asymptotically Optimal Balloon Density Estimates

Given a sample of n observations from a density ƒ on R d , a natural estimator of ƒ( x) is formed by counting the number of points in some region R surrounding x and dividing this count by the d dimensional volume of R . This paper presents an asymptotically optimal choice for R . The optimal shape turns out to be an ellipsoid, with shape depending on x. An extension of the idea that uses a kernel function to put greater weight on points nearer x is given. Among nonnegative kernels, the familiar Bartlett-Epanechnikov kernel used with an ellipsoidal region is optimal. When using higher order kernels, the optimal region shapes are related to L p balls for even positive integers p.

Criteria of General Weak Type Inequalities for Integral Transforms with Positive Kernels

Abstract Necessary and sufficient conditions are derived in order that an inequality of the form be fulfilled for some positive c independent of λ and a ν-measurable nonnegative function ƒ : X → R 1, where k : X × X × [0, ∞) → R 1 is a nonnegative measurable kernel, (X, d, μ) is a homogeneous type space, φη and ψ are quasiconvex functions, ψ ∈ Δ2, and t –α θ(t) is a decreasing function for some α, 0 < α < 1. A similar problem was solved in Lorentz spaces with weights.

Asymptotically best bandwidth selectors in kernel density estimation

This paper gives asymptotically best data based choices of the bandwidth of the kernel density estimator. These bandwith selectors attain the fastest possible rate of convergence to the desired theoretical optimum and the best possible constant coefficient in the spirit of the usual Fisher Information, with the use of only nonnegative kernel estimators at all stages of the selection process.

A note on bias reduction in variable‐kernel density estimates

AbstractIt is well known that the inverse‐square‐root rule of Abramson (1982) for the bandwidth h of a variable‐kernel density estimator achieves a reduction in bias from the fixed‐bandwidth estimator, even when a nonnegative kernel is used. Without some form of “clipping” device similar to that of Abramson, the asymptotic bias can be much greater than O(h4) for target densities like the normal (Terrell and Scott 1992) or even compactly supported densities. However, Abramson used a nonsmooth clipping procedure intended for pointwise estimation. Instead, we propose a smoothly clipped estimator and establish a globally valid, uniformly convergent bias expansion for densities with uniformly continuous fourth derivatives. The main result extends Hall's (1990) formula (see also Terrell and Scott 1992) to several dimensions, and actually to a very general class of estimators. By allowing a clipping parameter to vary with the bandwidth, the usual O(h4) bias expression holds uniformly on any set where the target density is bounded away from zero.

No Arbitrage and Arbitrage Pricing: A New Approach

ABSTRACTWe argue that arbitrage‐pricing theories (APT) imply the existence of a low‐dimensional nonnegative nonlinear pricing kernel. In contrast to standard constructs of the APT, we do not assume a linear factor structure on the payoffs. This allows us to price both primitive and derivative securities. Semi‐nonparametric techniques are used to estimate the pricing kernel and test the theory. Empirical results using size‐based portfolio returns and yields on bonds reject the nested capital asset‐pricing model and linear APT and support the nonlinear APT. Diagnostics show that the nonlinear model is more capable of explaining variations in small firm returns.

Towards statistics of crystal orientations in quantitative texture anaylsis

By the application of Rodrigues parameters, crystal orientations are represented as points on the unit semi-hypersphere S4+ ⊃ \bb R4 or equivalently on the projective hyperplane H3 ⊃ \bb R4. For the statistical analysis of orientation data, probability models on S4+ ≡ H3 are required. Among different hyperspherical analogs of the normal distribution in Euclidean space, corresponding to various of its characterizations, the Bingham model distribution is characterized as the hyperspherical analog for statistical purposes. Maximum-likelihood estimates of its parameter matrices and single-sample significance tests of uniformity and rotational symmetry are presented. In terms of probability theory and stochastic processes, the hyperspherical Brownian-diffusion distribution is the analog of the normal distribution in Euclidean space. An orientation distribution derived by Savelova [Ind. Lab. (USSR), (1984), 50, 468–474] and erroneously related to a central-limit-type theorem for rotations by Matthies, Muller & Vinel [Textures Microstruct. (1988), 10, 77–96], with the Brownian-diffusion distribution on S4+ ≡ H3, recalls the fact that no simple analog of the central limit theorem in Euclidean space exists for arbitrary spaces and manifolds, e.g. for SP, SP+, SO(p), p ≥ 2. Therefore, interpretations of preferred orientations in terms of mechanisms of texture development related to an imaginary simple analog of the central limit theorem would generally be misleading. Individual crystal-orientation measurements may initially require different nonparametric methods of analysis. As a counterpart of pole-to-orientation density inversion of diffraction data, hyperspherical kernel orientation density estimation is suggested for individual orientation measurements. With the application of non-negative kernels, the estimated orientation density is always non-negative. The critical smoothing parameter of this method is determined on information-theoretical grounds. For patterns of preferred orientation too complex to be sufficiently approximated by the parametric Bingham model distribution, a clustering of individual orientation measurements into disjunct classes is suggested that relates similarity of orientations to multimodality of the estimated orientation density; each class of orientations may then be individually analyzed by parametric methods.

Spectral inequalities for compact integral operators on Banach function spaces

AbstractThis article generalizes some spectral inequalities for non-negative matrices (see [2] and [3]) to compact integral operators with non-negative kernels defined on Banach function spaces. The spectral radius of a sum of such operators is estimated under certain conditions and a generalization of this inequality is given. An inequality for the spectral radius of a compact integral operator with the kernel equal to a weighted geometric mean of non-negative kernels is also proved.

Nuclearity of a nonnegative definite integral kernel on a separable metric space

The aim of this paper is to characterize the nuclearity of an integral operator, defined by a continuous non-negative definite square integrable kernel on a separable metric space, in terms of the integrability of the trace of the kernel function. Nuclearity here plays a role forU-statistics.

On nonnegative solvability of linear integral equations

Let (Ω,Σ,μ) denote a σ-finite measure space, and L p (Ω,Σ,μ) (1⩽ p<∞) the usual Banach lattices of pth summable real-valued functions. Suppose, moreover, K is an integral operator whose nonnegative kernel k(·,·) is ( Σ× Σ)-measurable on Ω×Ω and which maps L p (Ω,Σ,μ) into itself while possessing a compact iterate. We present necessary and sufficient conditions for the integral operator equation λƒ = Kƒ+g to possess a nonnegative solution ƒϵ L p (Ω,Σ,μ) whenever g is a given nontrival and nonnegative element of L p (Ω,Σ,μ) and λ is any given positive parameter. This analysis extends that by Victory [ SIAM J. Algebraic Discrete Methods 6:406–412 (1985)] for the matrix case.

On a Theorem of Janson

Previous article Next article On a Theorem of JansonV. G. MikhailovV. G. Mikhailovhttps://doi.org/10.1137/1136018PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Svante Janson, Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs, Ann. Probab., 16 (1988), 305–312 89a:60062 0639.60029 CrossrefGoogle Scholar[2] M. V. Petrovskaya and , A. M. Leontovich, The central limit theorem for a sequence of random variables with a slowly growing number of dependencies, Theory Probab. Appl., 27 (1982), 815–825 10.1137/1127089 0559.60069 LinkGoogle Scholar[3] A. M. Leontovich, On a condition for slowly growing number of dependencies, Theory Probab. Appl., 30 (1985), 196–200 10.1137/1130026 0657.60015 LinkGoogle Scholar[4] V. G. Mikhailov, On the asymptotic normality of U-statistics with non-negative kernels, Proc. of the Steklov Math. Inst., 1988, 105–113, issue 4 Google Scholar[5] V. P. Leonov and , A. N. Shiryaev, On a method of semi-invariants, Theor. Probability Appl., 4 (1959), 319–329 23:A673 0087.33701 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Об условиях асимптотической нормальности числа повторений в стационарной случайной последовательности28 August 2021 | Дискретная математика, Vol. 33, No. 3 Cross Ref Central limit theorems for patterns in multiset permutations and set partitionsThe Annals of Applied Probability, Vol. 30, No. 1 Cross Ref Weighted dependency graphsElectronic Journal of Probability, Vol. 23, No. none Cross Ref Dependency graphs and mod-Gaussian convergence7 December 2016 Cross Ref Asymptotic normality of the number of values of m-dependent random variables which occur a given number of timesDiscrete Mathematics and Applications, Vol. 21, No. 1 Cross Ref Asymptotic normality of the number of absent noncontinuous chains of outcomes of independent trialsDiscrete Mathematics and Applications, Vol. 19, No. 3 Cross Ref A Refinement of the Central Limit Theorem for Sums of Independent Random IndicatorsA. Yu. Volkova12 July 2006 | Theory of Probability & Its Applications, Vol. 40, No. 4AbstractPDF (442 KB) Volume 36, Issue 1| 1992Theory of Probability & Its Applications History Submitted:30 January 1989Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1136018Article page range:pp. 173-176ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

A Simple Root $n$ Bandwidth Selector

The asymptotically best bandwidth selectors for a kernel density estimator currently require the use of either unappealing higher order kernel pilot estimators or related Fourier transform methods. The point of this paper is to present a methodology which allows the fastest possible rate of convergence with the use of only nonnegative kernel estimators at all stages of the selection process. The essential idea is derived through careful study of factorizations of the pilot bandwidth in terms of the original bandwidth.

On correcting for variance inflation in kernel density estimation

It is a simple matter to correct for the well-known variance inflation property of nonnegative kernel density estimates whereby the estimated distribution's variance exceeds that of the sample. But should we bother? Asymptotic mean integrated squared error considerations, developed here for the first time, suggest we may. However, we observe that the difference variance correction makes is, in most practical instances, negligible. Even when this is not so, exploratory conclusions would rarely be affected and, on occasions when this is not so either, variance correction can have a slight tendency to obscure potentially important features of the density. An exception to all this is estimation of the normal density for which correcting for variance inflation is certainty appropriate. This author retains a personal preference for continuing with uncorrected kernel density estimates, but the main message of the paper is the relative indifference to whether or not variance correction is employed.

On the Asymptotic Distributions of Bandwidth Estimates

The problem of automatic bandwidth selection for a kernel regression estimate is studied. Since the bandwidth considerably affects the features of the estimated curve, it is important to understand the behavior of bandwidth selection procedures. The bandwidth estimate considered here is the minimizer of Mallows' criterion. Though it was established that the bandwidth estimate is asymptotically normal, it is well recognized that the rate of convergence is extremely slow. In simulation studies, it is often observed that the normal distribution does not provide a satisfactory approximation. In this paper, the bandwidth estimate is shown to be approximately equal to a constant plus a linear combination of independent exponential random variables. In practice, the distribution of a weighted sum of chi-squared random variables can be approximated by a multiple of a chi-squared distribution. Simulation results indicate that this provides a very good approximation even for a modest sample size. It is shown that the degrees of freedom of the chi-squared distribution goes to infinity rather slowly (at the rate $T^{1/5}$ for a nonnegative kernel). This explains why the distributions of the bandwidth estimates converge to the normal distribution so slowly.

Travelling Wave Solutions of a Nonlinear Diffusion Equation with Integral Term

Travelling wave solutions with compact support are investigated for a class of nonlinear nonlocal degenerate diffusion equations introduced by T. Nagai and M. Mimura. The problem is reduced to a boundary value problem for an integro-differential equation of second order and in turn to an integral equation with nonnegative kernel. There are given existene proofs for three types of power nonlinearities in case of a general integral term and closed solutions for a simple integral term.

A uniqueness theorem for the coagulation-fragmentation equation

This paper presents a uniqueness result for solutions to the general nonlinear coagulation-fragmentation equationwhereEquation (1·1) has many applications in the applied sciences (cf. [1, 3, 8, 13, 15]) and a brief physical interpretation can be found in Melzak [12] or the survey article by Drake[7]. c(x, t), for x ≥ 0, t ≥ 0, denotes the number of particles of size x at time t and the non-negative kernels K and F describe, respectively, the rates at which particles of size x coalesce with those of size y and particles of size (x + y) break-up into those of sizes x and y.

ASYMPTOTICS OF THE SOLUTION OF A MIXED PROBLEM FOR A SYSTEM OF DIFFERENTIAL EQUATIONS CONNECTED WITH A POSITIVE INTEGRAL OPERATOR

A mixed problem is considered for a hyperbolic system of partial differential equations for whose solutions there exist integral representations having a probability-theoretical meaning. It is assumed that the right-hand sides of the system satisfy exponential estimates. Under natural restrictions on the coefficients, the asymptotics of the solutions as t→∞ are obtained with the help of the theory of linear integral operators with nonnegative kernels, and the theory of branching processes with several types of particles in motion along an interval. Bibliography: 10 titles.

Nonmonotone Clines in Homogeneous Space Cannot Be Stable

This paper deals with models for the space-time variation of the genetic composition of populations. Both the standard reaction-diffusion equation and two nonlinear integral operator models are discussed. A simple argument is presented, explaining that nonconstant stationary solutions, so-called clines, in a homogeneous space cannot be stable unless they are monotone. In the integral operator case the proof uses differentiation along the orbits of the translation group and the Perron–Frobenius Theorem for integral operators with nonnegative kernel. For the reaction-diffusion equation stronger results are known, and only a heuristic argument is given.