Nonnegative Kernel Research Articles

Weighted Estimates for the Integral Operators with Monotone Kernel on a Half-Axis

We give criteria for the boundedness of integral operators with nonnegative monotone kernel in the weighted Lebesgue spaces on a half-axis.

EWMA Schemes with Nonhomogeneous Transition Kernels

In the last decades, various methods are applied for evaluating different performance measures of change-point detection schemes. In only some cases—recent examples are Chandrasekaran et al. (Chandrasekaran, S.; English, J.R.; Disney, R.L. Modeling and analysis of EWMA control schemes with variance-adjusted control limits. IIE Transactions 1995, 277, 282–290) and Steiner (Steiner, S.H. EWMA control charts with time-varying control limits and fast initial response. Journal of Quality Technology 1999, 31 (1), 75–86)—EWMA schemes with varying control limits are considered. However, EWMA charts with just these limits are sometimes more appropriate than those with fixed limits. Here, a computational approach is presented which allows to compute the usual performance measures with high precision. The main idea is connected to earlier results of Madsen and Conn (Madsen, R.W.; Conn, P.S. Ergodic behavior for nonnegative kernels. Ann. Probab. 1973, 1, 995–1013), Woodall (Woodall, W.H. The distribution of the run length of one-sided CUSUM procedures for continuous random variables. Technometrics 1983, 25, 295–301), and Waldmann (Waldmann, K.-H. Bounds for the distribution of the run length of geometric moving average charts. J. R. Stat. Soc., Ser. C, Appl. Stat. 1986a, 35, 151–158). Additionally, quantities as the steady-state ARL and the steady-state distribution of the chart statistic can be computed very precisely.

Expansion of a Self-Adjoint Absolutely Continuous Singular Integral Operator in Generalized Eigenvectors and Its Diagonalization

We describe the relationship between the expansion of a self-adjoint operator in generalized eigenvectors and the direct integral of Hilbert spaces. We perform the explicit diagonalization of a self-adjoint absolutely continuous singular integral operator Y using an Hermitian nonnegative kernel consisting of boundary values of the determining function of the operator T = X + iY with respect to the resolvent of the imaginary part of Y.

Global asymptotic behaviour in some functional parabolic equations

The global asymptotic behavior in some functional parabolic equations was presented. The sufficient condition for the global convergence to a spatially uniform equilibrium in the system of functional partial differential equations was provided. Results were observed to be valid for all non-negative spatio-temporal kernels, which also satisfied a uniform normailzation condition.

Poisson limits for U-statistics

We study Poisson limits for U-statistics with non-negative kernels. The limit theory is derived from the Poisson convergence of suitable point processes of U-statistics structure. We apply these results to derive infinite variance stable limits for U-statistics with a regularly varying kernel and to determine the index of regular variation of the left tail of the kernel. The latter is known as correlation dimension. We use the point process convergence to study the asymptotic behavior of some standard estimators of this dimension.

On the Construction and Frequency Localization of Finite Orthogonal Quadrature Filters

We introduce a new method to construct finite orthogonal quadrature filters using convolution kernels and show that every filter with value 1 at the origin can be obtained from an even nonnegative kernel. We apply the method to estimate the optimal frequency localization of finite filters. The frequency localization γp of a finite filter m0 is given by the distance in Lp-norm between |m0|2 and the Shannon low-pass filter. For each N>0 there is a filter mN0 of length 2N minimizing the value of γp. We prove that for such a minimizing sequence we have γpp(mN0)=O(1/N), 1⩽p⩽2, and this estimate is optimal. We construct several new families of both MRA and non-MRA filters with optimal asymptotic frequency localization.

Ratio Limit Theorems for Self-Adjoint Operators and Symmetric Markov Chains

A simplest ratio limit theorem is obtained for self-adjoint operators in the spaces of L2 type which leave invariant a cone of nonnegative elements. By means of the theorem we establish ratio limit theorems for symmetric Markov chains and symmetric kernels in measurable spaces. In particular, it is shown that for symmetric Harris recurrent Markov chains a result is valid which is an analogue of the known Orey theorem (1961) about discrete recurrent symmetric chains. Similar statements are valid for nonnegative symmetric quasi-Feller kernels on locally compact spaces which are Liouville in a certain sense.

Immanants and Finite Point Processes

Given a Hermitian, non-negative definite kernel K and a character χ of the symmetric group on n letters, define the corresponding immanant function Kχ[x1, …, xn]≔∑σχ(σ)∏ni=1K(xi, xσ(i)), where the sum is over all permutations σ of {1, …, n}. When χ is the sign character (resp. the trivial character), then Kχ is a determinant (resp. permanent). The function Kχ is symmetric and non-negative, and, under suitable conditions, is also non-trivial and integrable with respect to the product measure μ⊗n for a given measure μ. In this case, Kχ can be normalised to be a symmetric probability density. The determinantal and permanental cases or this construction correspond to the fermion and boson point processes which have been studied extensively in the literature. The case where K gives rise to an orthogonal projection of L2(μ) onto a finite-dimensional subspace is studied here in detail. The determinantal instance of this special case has a substantial literature because of its role in several problems in mathematical physics, particularly as the distribution of eigenvalues for various models of random matrices. The representation theory of the symmetric group is used to compute the normalisation constant and identify the kth-order marginal densities for 1⩽k⩽n as linear combinations of analogously defined immanantal densities. Connections with inequalities for immanants, particularly the permanental dominance conjecture of Lieb, are considered, and asymptotics when the dimension of the subspace goes to infinity are presented.

A Weiss-type model of ferromagnetism

The classical Weiss model of ferromagnetism is extended to nonuniform systems, by assuming the molecular field to be proportional to the convolution of the magnetization field with a nonnegative (e.g. Gaussian) kernel. Dealing with processes, a dynamic term accounting for dry friction-like dissipation due to wall pinning is also included in the constitutive equation. These models are mathematically well-behaved when coupled either with the magnetostatic equations or with Maxwell's equations. A similar modification is introduced for antiferromagnetism and ferrimagnetism.

Moser's second one-dimensional inequality for a class of integral operators

Suppose that \(1 < p < \infty \), \(q=p/(p-1)\), and for non-negative \(f\in L^p(-\infty\! ,\infty )\) and any real x we let \(F(x)-F(0)=\int _0^xf(t)\ dt\); suppose in addition that \(\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0\). Moser's second one-dimensional inequality states that there is a constant \(C_p\), such that \(\int\limits _{-\infty }^\infty \exp [a |F(x)|^q-|x|] \ dx\le C_p\) for each f with \(||f||_p\le 1\) and every \(a\le 1\). Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a.

Beta kernel estimators for density functions

Kernel estimators using non-negative kernels are considered to estimate probability density functions with compact supports. The kernels are chosen from a family of beta densities. The beta kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The proposed beta kernel estimators have two features. One is that the different amount of smoothing is allocated by naturally varying kernel shape without explicitly changing the value of the smoothing bandwidth. Another feature is that the support of the beta kernels can match the support of the density function; this leads to larger effective sample sizes used in the density estimation and can produce density estimates that have smaller finite-sample variance than some other estimators.

Multivariate Boundary Kernels and a Continuous Least Squares Principle

Summary Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean-squared error or sup-norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non-negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel-generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss–Markov theorem.

Loewner’s theorem for kernels having a finite number of negative squares

By a theorem of Loewner, a continuously differentiable real-valued function on a real interval whose difference quotient is a nonnegative kernel is the restriction of a holomorphic function which has nonnegative imaginary part in the upper half-plane and is holomorphic across the interval. An analogous result is obtained when the difference-quotient kernel has a finite number of negative squares.

Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for Nonnegative and symmetrized U-Statistics

This paper concerns moment and tail probability inequalities and the strong law of large numbers for $U$-statistics with nonnegative or symmetrized kernels and their multisample and decoupled versions. Sub-Bernoulli functions are used to obtain the moment and tail probability inequalities, which are then used to obtain necessary and sufficient conditions for the almost sure convergence to zero of normalized $U$-statistics with nonnegative or completely symmetrized kernels, without further regularity conditions on the kernel or the distribution of the population, for normalizing constants satisfying a simple condition. Moments of $U$-statistics are bounded from above and below by that of maxima of certain kernels, up to scaling constants. The multisample and decoupled versions of these results are also considered.

Iterative algorithms for deblurring and deconvolution with constraints

The deblurring problem is that of recovering the function from (noisy) values . The discrete finite version of the problem is to solve the system of linear equations Hc=d for c, where H is a matrix and d and c are vectors. When the kernel is a function of the difference , the deblurring problem becomes a deconvolution problem. The use of iterative algorithms to effect deblurring subject to non-negativity constraints on c has been presented by Snyder et al for the case of non-negative kernel function h. In this paper we extend these algorithms to include upper and lower bounds on the entries of the desired solution. We show that any linear deblurring problem involving a real kernel h can be transformed into a linear deblurring problem involving a non-negative kernel. Therefore our algorithms apply to general deblurring and deconvolution problems. These algorithms converge to a solution of the system of equations y = Px, with , for , , satisfying the vector inequalities , whenever such a solution exists. When there is no solution satisfying the constraints the simultaneous versions converge to an approximate solution that minimizes a cost function related to the Kullback-Leibler cross-entropy and the Fermi-Dirac generalized entropy.

On the Schur Test forL2-Boundedness of Positive Integral Operators with a Wiener–Hopf Example

The Schur sufficiency condition for boundedness of any integral operator with non-negative kernel betweenL2-spaces is deduced from an observation, Proposition 1.2, about the central role played byL2-spaces in the general theory of these operators. Suppose (Ω,M,μ) is a measure space and thatK:Ω×Ω→[0,∞) is an M×M-measurable kernel. The special case of Proposition 1.2 for symmetrical kernels says that such a linear integral operator is bounded onanyreasonable normed linear spaceXof M-measurable functions only if it is bounded onL2(Ω,M,μ) where its norm is no larger. The general form of Schur's condition (Halmos and Sunder “Bounded Integral Operators onL2-Spaces,” Springer-Verlag, Berlin/New York, 1978) is a simple corollary which, in the symmetrical case, says that the existence of an M-measurable (not necessarily square-integrable) functionh>0μ-almost-everywhere onΩwithKh(x)=∫ΩK(x,y)h(y)μ(dy)⩽Λh(x)(x∈Ω) (*)implies thatKis a bounded (self-adjoint) operator onL2(Ω,M,μ) of norm at mostΛ. When (Ω,M,μ) isσ-finite, we show that Schur's condition is sharp: in the symmetrical case the boundedness of K onL2(Ω,M,μ) implies, for anyΛ>‖K‖2, the existence of a functionh∈L2(Ω,M,μ) which is positiveμ-almost-everywhere and satisfies (*). Such functionshsatisfying (*), whether inL2(Ω,M,μ) or not, will be calledSchur test functions. They can be found explicitly in significant examples to yield best-possible estimates of the norms for classes of integral operators with non-negative kernels. In the general theory the operators are not required to be symmetrical (a theorem of Chisholm and Everitt (Proc. Roy. Soc. Edinburgh Sect. A69(14) (1970/1971), 199–204) on non-self-adjoint operators is derived in this way). They may even act between differentL2-spaces. Section 2 is a rather substantial study of how this method yields the exact value of the norm of a particular operator between differentL2-spaces which arises naturally in Wiener–Hopf theory and which has several puzzling features.

On the basic solutions to the generalized fuzzy integral equation

For the fuzzy measure space ( X, A, μ), the structure of the basic solutions to the generalized fuzzy integral equation >. esh; x ( φ( x) ∧ h( x)) dμ = β is considered, where h( x) is taken as a nonnegative measurable kernel function, β ϵ(0, ∞) is a constant, ∧ denotes the logic multiplication (minimum operator). At first, the existence of the constant basic solution to Eq. (1) is investigated, and the solvable sufficient and necessary conditions to the Eq. (1) are characterized, then the characteristic functional basic solution to Eq. (1) is shown. In addition, two convergent theorems corresponding to the sequences of approximate basic solutions each of which is constituted by the series with the logic addition (maximum operator denoted by ⋁) for the characteristic functions are obtained.

A simple root n bandwidth selector for nonparametric regression

The purpose of this paper is to investigate data-driven bandwidth selection for non-parametric regression based on a double-smoothing procedure. It will be shown that the best convergence rate can be achieved by kernel regression with non-negative kernels in both pilot smoothing and as well as in main smoothing. The asymptotic results are given for a naive kernel estimator with an equally spaced design, but they can also be used for other kernel estimators or for locally weighted regression. Three variates of data-driven bandwidth selectors for local linear regression are proposed. One of them, ĥ DSI, is root n consistent. The performance of these bandwidth selectors is studied through simulation. They are also compared with the bandwidths selected by the R criterion of Rice and the true ASE optimal bandwidth (h ASE). IN spite of satisfactory performances of all bandwidth selectors, the root n one turns out to be the best in theory as well as in practice.

Weighted weak-type inequalities for the maximal function of nonnegative integral transforms over approach regions

The relation between approach regions and singularities of nonnegative kernels Kt(x,y) is studied, where t E (O,oo), x, y E X, and X is a homogeneous space. For 1 < p < q < oc, a sufficient condition on approach regions Qa (a E X) is given so that the maximal function sup f Kt(x, y)f(y) du(y) (X,t) E aX is weak-type (p, q) with respect to a pair of measures a and w. It is shown that this condition is also necessary for operators of potential type in the sense of Sawyer and Wheedon (Amer. J. Math. 114 (1992), 813-874).

On the root- N-consistent semiparametric estimation of partially linear models

In the context of a partially linear model, we show that a √ n-consistent estimator for the coefficients of the parametric part of the regression function can be obtained by using a non-negative second-order kernel function as long as the dimension of the variables in the non-parametric part is less than or equal to, five.