A note on the performance of the gamma kernel estimators at the boundary

The gamma kernel estimator is proposed in Chen [Chen, S.X., 2000. Probability density function estimation using gamma kernels. Annals of the Institute of Statistical Mathematics 52, 471–480] to estimate densities with support $[0,\infty )$. It is shown in his paper that the gamma kernel estimator is non-negative, free of boundary bias, and achieves the optimal rate of convergence for the mean integrated squared error. Numerical results reported in Chen’s paper show that, in the boundary region, the gamma kernel estimator even outperforms some widely used boundary corrected density estimators such as the boundary kernel estimator. However, our study finds that the gamma kernel estimator at $x=0$ is actually the reflection estimator when the double exponential kernel is used and is only boundary problem free when the estimated density has a shoulder at $x=0$ (i.e., the first derivative of the density at $x=0$ is zero). For densities not satisfying the shoulder condition, we show that the gamma kernel estimator has a severe boundary problem and its performance is inferior to that of the boundary kernel estimator.     

Mass shifting roles of negative kernel mass in density estimation

f(x) is a univariate density in C ⁴ with bounded support. For any n and sufficiently small kernel bandwidths, the symmetric appendage of any negative mass, –U, to any smooth unimodal symmetric kernel of order p=2 shifts expected estimator mass from regions where f(x)>0 to regions where f(x)<0. For large n, the mean automatic kernel adaptation induced by –U is analyzed in the simplest MISE reduction scenario: The symmetric appendage of –U to the uniform kernel K(x, X) over MISE-optimal bandwidths reduces MISE by shifting K(x, X) mass asymmetrically across the observation X in the direction of decreasing |f(x)|.     

Variable location and scale kernel density estimation

Variable (bandwidth) kernel density estimation (Abramson (1982,Ann. Statist.,10, 1217–1223)) and a kernel estimator with varying locations (Samiuddin and El-Sayyad (1990,Biometrika,77, 865–874)) are complementary ideas which essentially both afford bias of orderh ⁴ as the overall smoothing parameterh 0, sufficient differentiability of the density permitting. These ideas are put in a more general framework in this paper. This enables us to describe a variety of ways in which scale and location variation may be extended and/or combined to good theoretical effect. This particularly includes extending the basic ideas to provide new kernel estimators with bias of orderh ⁶. Technical difficulties associated with potentially overly large variations are fully accounted for in our theory.     

Positive definite dot product kernels in learning theory

In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=xy are used. In many models of learning theory, polynomial kernels K(x,y)=_l=0^Na_l(xy)^l generating polynomials of degree N, and dot product kernels K(x,y)=_l=0⁺a_l(xy)^l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f:RR such that the matrix [f(xⁱx^j)]_i,j=1^m is positive semi-definite for any x¹,x²,...,x^mRⁿ, n2. Supported by CERG Grant No. CityU 1144/01P and City University of Hong Kong Grant No. 7001342.AMS subject classification 42A82, 41A05     

Deconvolving a density from contaminated dependent observations

The paper studies the performance of deconvoluting kernel density estimators for estimating the marginal density of a linear process. The data stem from the linear process and are partially, respectively fully contaminated by iid errors with a known distribution. If 1–p denotes the proportion of contaminated observations (and it is, of course, unknown which observations are contaminated and which are not) then for 1–p (0, 1) and under mild conditions almost sure deconvolution rates of orderO(n ^–2/5(logn)^9/10) can be achieved for convergence in . This rate compares well with the existing rates foriid uncontaminated observations. Forp=0 and exponentially decreasing error characteristic function the corresponding rates are of merely logarithmic order. As a by-product the paper also gives a rate of convergence result for the empirical characteristic function in the linear process context and utilizes this to demonstrate that deconvoluting kernel density estimators attain the optimal rate in the dependence case with exponentially decreasing error characteristic function.This work was partially supported by a grant from the Deutsche Forschungsgemeinschaft.     

Additive Kernels and Integral Representation of Potentials

Let P=(P _t ) _t>0 be a submarkovian semigroup of kernels on a measurable space (X,). An additive kernel of P is a kernel K from X into ]0,[ such that P _t K(x,A)=K(x,A+t) for every t>0,xX and every Borel subset A of ]0,[. It is proved in this paper that for every potential f of P, there exits an additive kernel K of P, unique (up to equivalence) such that f=K1=₀ K(,dt). This result is already well known for regular potentials of right processes. If U=(U _p ) _p>0 is a sub-Markovian resolvent of kernels on (X,), we give a notion of additive kernel of U and we prove a similar integral representation of potentials of U.     

On the order of the error in discretization methods for weakly singular second kind non-smooth solutions

In general, second kind Volterra integral equations with weakly singular kernels of the formk(t,s)(t –s)^– posses solutions which have discontinuous derivatives att=0. A discrete Gronwall inequality is employed to prove that, away from the origin, the error in product integration and collocation schemes for these equations is of order 2-.     

Computing a family of reproducing kernels for statistical applications

For an open subset of , an integer,m, and a positive real parameter , the Sobolev spacesH ^m () equipped with the norms: u²=u(t)²dt+(1/^2m u ^(m)(t)² constitute a family of reproducing kernel Hilbert spaces. When is an open interval of the real line, we describe the computation of their reproducing kernels. We derive explicit formulas for these kernels for all values ofm in the case of the whole real line, and form=1 andm=2 in the case of a bounded open interval.This research was partly supported by NSF Grant DMS-9002566.     

A global existence theorem for the general coagulation–fragmentation equation with unbounded kernels

In this article an existence theorem is proved for the coagulation–fragmentation equation with unbounded kernel rates. Solutions are shown to be in the space X⁺ = {c∈L¹: ∫ (1 + x)∣c(x)∣dx < ∞} whenever the kernels satisfy certain growth properties and the non-negative initial data belong to X⁺. The proof is based on weak L¹ compactness methods applied to suitably chosen approximating equations.     

Asymptotic behavior of large solutions to -Laplacian of Bieberbach–Rademacher type

By the Karamata regular variation theory and the method of lower and upper solutions, we establish the asymptotic behavior of boundary blow-up solutions of the quasilinear elliptic equation div(|u|^p−2u)=b(x)f(u) in a bounded ΩR^N subject to the singular boundary condition u(x)=∞, where the weight b(x) is non-negative and non-trivial in Ω, which may be vanishing on the boundary or go to unbounded, the nonlinear term f is a Γ-varying function at infinity, whose variation at infinity is not regular.     

An Ω-theorem...: Addendum

It is proved in [1] that the error term associated with the summatory function of the divisor function _–1 (n) satisfies (1)E _–1 (loglogx) with an implied constantC _–(e –1)/2. In fact, the method of [1] yields the better (2)C _–e /2, and also (3)E _–1 (log logx) with an implied constant (4)C ₊e /2.     

Asymptotic bias and variance for a general class of varying bandwidth density estimators

Summary We consider a general class of varying bandwidth estimators of a probability density function. The class includes the Abramson estimator, transformation kernel density estimator (TKDE), Jones transformation kernel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-Linton-Nielsen estimator (JLN), Taylor series approximations of TKDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinary kernel estimator , it is possible to iterate and compute a sequence of estimates , using each estimate as a pilot estimator in the next step. The first main result is a formula for the bias order. If the bandwidths used in different steps have a common orderh=h(n), the bias of is of orderh ^2km ,k=1, ...,t. Hereh ^m is the bias order of the ideal estimator (defined by using the unknownf as pilot). The second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates. Ifm<, it is possible to make asymptotically equivalent to the ideal estimator.     

Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Let X, X ₁, X ₂,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F _n the distribution function of centered and normed sum S _n . Let F belong to the domain of attraction of the standard normal law , that is, lim F _n (x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx ^––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n ^–1/2) and then add new terms of orders n ^–/2 ln n, n ^–/2 ln^-1 n, etc., where 0.     

An O(n 2) Active Set Algorithm for Solving Two Related Box Constrained Parametric Quadratic Programs

Recently, O(n ²) active set methods have been presented for minimizing the parametric quadratic functions (1/2)x Dx–a x+| x–c| and (1/2)x Dx–a x+(/2)( x–c)², respectively, subject to lxb, for all nonnegative values of the parameter . Here, D is a positive diagonal n×n matrix, and a are arbitrary n-vectors, c is an arbitrary scalar; l and b are arbitrary n-vectors such that lb. In this paper, we show that each one of these algorithms may be used to simultaneously solve both parametric programs withno additional computational cost.     

Kernel B-splines and interpolation

This paper applies difference operators to conditionally positive definite kernels in order to generate kernel -splines that have fast decay towards infinity. Interpolation by these new kernels provides better condition of the linear system, while the kernel -spline inherits the approximation orders from its native kernel. We proceed in two different ways: either the kernel -spline is constructed adaptively on the data knot set , or we use a fixed difference scheme and shift its associated kernel -spline around. In the latter case, the kernel -spline so obtained is strictly positive in general. Furthermore, special kernel -splines obtained by hexagonal second finite differences of multiquadrics are studied in more detail. We give suggestions in order to get a consistent improvement of the condition of the interpolation matrix in applications.     

Radial basis interpolation on homogeneous manifolds: convergence rates

Pointwise error estimates for approximation on compact homogeneous manifolds using radial kernels are presented. For a positive definite kernel κ the pointwise error at x for interpolation by translates of κ goes to 0 like ρ ^r , where ρ is the density of the interpolating set on a fixed neighbourhood of x. Tangent space techniques are used to lift the problem from the manifold to Euclidean space, where methods for proving such error estimates are well established. Partially supported by NSF Grant DMS-9972004.     

Aircraft control for flight in an uncertain environment: Takeoff in windshear

The design of the control of an aircraft encountering windshear after takeoff is treated as a problem of stabilizing the climb rate about a desired value of the climb rate. The resulting controller is a feedback one utilizing only climb rate information. Its robustness vis-a-vis windshear structure and intensity is illustrated via simulations employing four different windshear models.Notations ARL aircraft reference line - D drag force, lb - g gravitational force per unit mass=const, ft sec^–2 - h vertical coordinate of aircraft center of mass (altitude), ft - L lift force, lb - m aircraft mass=const, lb ft^–1 sec² - O mass center of aircraft - S reference surface, ft² - t time, sec - T thrust force, lb - V aircraft speed relative to wind-based reference frame, ft sec^–1 - V _e aircraft speed relative to ground, ft sec^–1 - W _x horizontal component of wind velocity, ft sec^–1 - W _h vertical component of wind velocity, ft sec^–1 - x horizontal coordinate of aircraft center of mass, ft - relative angle of attack, rad - relative path inclination, rad - _e path inclination, rad - thrust inclination, rad - air density=const, lb ft² sec² Dot denotes time derivative.     

Some analytical properties of γ-convex functions on the real line

This paper deals with the analytical properties of -convex functions, which are defined as those functions satisfying the inequalityf(x ₁ )+f(x ₂ )f(x ₁)+f(x ₂), forx _i [x ₁,x ₂], |x _i –x _i |=, i=1,2, whenever |x ₁–x ₂|>, for some given positive . This class contains all convex functions and all periodic functions with period . In general, -convex functions do not have ideal properties as convex functions. For instance, there exist -convex functions which are totally discontinuous or not locally bounded. But -convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length . It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of -convex functions on the real line. However, -convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of -convexity are given. Ways for generating and representing -convex functions are described.This research was supported by the Deutsche Forschungsgemeinschaft. The first author thanks Prof. Dr. E. Zeidler and Prof. Dr. H. G. Bock for their hospitality and valuable support.     

Transportation cost for Gaussian and other product measures

Consider the canonical Gaussian measure _N on , a probability measure on ^N , absolutely continuous with respect to _N . We prove that the transportation cost of to _N , when the cost of transporting a unit of mass fromx toy is measured by x–y², is at most d. As a consequence we obtain a completely elementary proof of a very sharp form of the concentration of measure phenomenon in Gauss space. We then prove a result of the same nature when _N is replaced by the measure of density 2^–N exp (– _iN |x _i |). This yields a sharp form of concentration of measure in that space.Work partially supported by an NSF Grant.     

Theq-gamma function forx<0

F. H. Jackson defined aq analogue of the gamma function which extends theq-factorial (n!) _q =1(1+q)(1+q+q ²)...(1+q+q ²+...+q ^n–1) to positivex. Askey studied this function and obtained analogues of most of the classical facts about the gamma function, for 0<q<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfyingf(1)=1 andf(x+1)=[(q ^x –1)/(q–1)]f(x) is in fact theq-gamma function He also studied the behavior of _q asq changes and showed that asq1^–, theq-gamma function becomes the ordinary gamma function forx>0.I proved many of these results forq>1. The current paper contains a study of the behavior of _q (x) forx<0 and allq>0. In addition to some basic properties of _q , we will study the behavior of the sequence {x _n (q)} of critical points asn orq changes.