Asymptotic bounds for the expected L error of a multivariate kernel density estimator

The kernel estimator of a multivariate probability density function is studied. An asymptotic upper bound for the expected L¹ error of the estimator is derived. An asymptotic lower bound result and a formula for the exact asymptotic error are also given. The goodness of the smoothing parameter value derived by minimizing an explicit upper bound is examined in numerical simulations that consist of two different experiments. First, the L¹ error is estimated using numerical integration and, second, the effect of the choice of the smoothing parameter in discrimination tasks is studied.     

Bahadur representation of the kernel quantile estimator under truncated and censored data

1.IntroductionTheestimationofpopulationquaillesisofgrestillterestwhenone.isnotpreparedtoassumeaparametricformfortheunderlyingdistribution.Inaddition,quaillesoftenariseasthensturalthingtoestimatewhentheunderlyingdistributionisskewed.LetXIIXZ,')Xubei...     

Rate of convergence of a convolution-type estimator of the marginal density of a MA(1) process

In this paper moving-average processes with no parametric assumption on the error distribution are considered. A new convolution-type estimator of the marginal density of a MA(1) is presented. This estimator is closely related to some previous ones used to estimate the integrated squared density and has a structure similar to the ordinary kernel density estimator. For second-order kernels, the rate of convergence of this new estimator is investigated and the rate of the optimal bandwidth obtained. Under limit conditions on the smoothing parameter the convolution-type estimator is proved to be -consistent, which contrasts with the asymptotic behavior of the ordinary kernel density estimator, that is only -consistent.     

Robust model selection criteria for robust Liu estimator

In linear regression analysis, outliers often have large influence in the model/variable selection process. The aim of this study is to select the subsets of independent variables which explain dependent variables in the presence of multicollinearity, outliers and possible departures from the normality assumption of the error distribution in robust regression analysis. In this study to overcome this combined problem of multicollinearity and outliers, we suggest to use robust selection criterion with Liu and Liu-type M(LM) estimators.     

Multivariate approximation in the worst case setting over reproducing kernel Hilbert spaces

We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L_∞ norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper.We show that the L_∞ approximation problem in the worst case setting is related to the weighted L₂ approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter α>1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-(α-1)/2 and n^-1/2, respectively.We also study tractability of L_∞ approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L_∞ approximation in terms of tractability conditions of the weighted L₂ approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.     

Enumeration by kernel positions

We introduce a class of two-player games on posets with a rank function, in which each move of the winning strategy is unique. This allows one to enumerate the kernel positions by rank. The main example is a simple game on words in which the number of kernel positions of rank n is a signed factorial multiple of the nth Bernoulli number of the second kind. Generalizations to the degenerate Bernoulli numbers and to negative integer substitutions into the Bernoulli polynomials are developed. Using an appropriate scoring system for each function with an appropriate Newton expansion we construct a game in which the expected gain of a player equals the definite integral of the function on the interval [0,1].     

Adaptive nonparametric estimation of a multivariate regression function

We consider the kernel estimation of a multivariate regression function at a point. Theoretical choices of the bandwidth are possible for attaining minimum mean squared error or for local scaling, in the sense of asymptotic distribution. However, these choices are not available in practice. We follow the approach of Krieger and Pickands (Ann. Statist.9 (1981) 1066–1078) and Abramson (J. Multivariate Anal.12 (1982), 562–567) in constructing adaptive estimates after demonstrating the weak convergence of some error process. As consequences, efficient data-driven consistent estimation is feasible, and data-driven local scaling is also feasible. In the latter instance, nearest-neighbor-type estimates and variance-stabilizing estimates are obtained as special cases.     

Equivalence of the C*-Algebras qℂ and C 0(ℝ2) in the Asymptotic Category

The results of Kasparov, Connes, Higson, and Loring imply the coincidence of the functors [[qℂ ⊗ K, B ⊗ K]] = [[C ₀(ℝ²) ⊗ K, B ⊗ K]] for any C*-algebra B; here[[A, B]] denotes the set of homotopy classes of asymptotic homomorphisms from A to B. Inthe paper, this assertion is strengthened; namely, it is shown that the algebras qℂ ⊗ K and C ₀(ℝ²) ⊗ K are equivalent in the category whose objects are C*-algebras and morphisms are classes of homotopic asymptotic homomorphisms. Some geometric properties of the obtained equivalence are studied. Namely, the algebras qℂ ⊗ K and C ₀(ℝ²) ⊗ K are represented as fields of C*-algebras; it is proved that the equivalence is not fiber-preserving, i.e., is does not take fibers to fibers. It is also proved that the algebras under consideration are not homotopy equivalent.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 788–796.Original Russian Text Copyright ©2005 by T. V. Shul’man.     

Variable stepsize schemes for effective order methods and enhanced order composition methods

The interest in the concept of effective order has been revived by its rediscovery in applications to symplectic problems. In this paper we revert to the original application, the construction of explicit Runge–Kutta methods. Changing stepsize is a characteristic difficulty with effective order methods and we propose a way of overcoming this difficulty. We also consider the possible cancellation of local truncation errors of two methods over two successive steps. Using the algebraic approach for deriving these results gives us further insight into these methods and compositions of methods. A particular sixth stage Runge–Kutta pair is derived in the paper and is shown to be competitive.     

A rank estimator in the two-sample transformation model with randomly censored data

We consider the transformation model which is a generalization of Lehmann alternatives model. This model contains a parameter and a nonparametric part F ₁ which is a distribution function. We propose a kind of M-estimator of based on ranks in the presence of random censoring. It is nonparametric in the sense that we do not have to know F ₁. Moreover, it is simple and asymptotically normal. For the proportional hazards model with special censoring, we obtain the asymptotic relative efficiency of our estimator with respect to the best nonparametric estimator for this model. It is quite efficient for special values of . We also make a comparison between our estimator and other proposed estimators with real data.     

Effect of the initial estimator on the asymptotic behavior of one-step M-estimator

For a general (real) parameter, let M _nbe the M-estimator and M _n ⁽¹⁾ be its one-step version (based on a suitable initial estimator M _n ⁽⁰⁾). It is known that, under certain regularity conditions, n(M _n ⁽¹⁾-M _n)=O _p(1). The asymptotic distribution of n(M _n ⁽¹⁾-M _n) is studied; it is typically non-normal and it reveals the role of the initial estimator M _n ⁽⁰⁾.Work of this author was partially supported by the Office of Naval Research, Contract No. N00014-83-K-0387     

C-wellposedness of the complete second order abstract cauchy problem and applications

In this paper, we first give a sufficient and necessary condition for to generate an exponentially bounded -semigroup and discuss its relations to the C-wellposedness of the complete second order abstract Cauchy problem ((ACP₂) for short) in some sense. Then we use these results and those in [1] to discuss the C-(exponential) wellposedness of a kind of (ACP₂) with application backgrounds, and develop the results in [2]. This project is supported by the NNSF of China, and the Youth Science and Technique Foundation of Shanxi Province, China     

On the <Emphasis Type="Italic">C</Emphasis>*-valued triangle equality and inequality in Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

We prove that for two elements x, y in a Hilbert C*-module V over a C*-algebra the C*-valued triangle equality |x + y| = |x| + |y| holds if and only if 〈x, y〉 = |x| |y|. In addition, if has a unit e, then for every x, y ∊ V and every ɛ > 0 there are contractions u, υ ∊ such that |x + y| ≦ u|x|u* + υ|y|υ* + ɛe.      

The rate of convergence for the least squares estimator in nonlinear regression model with dependent errors

We study the parameter estimation in a nonlinear regression model with a general error's structure,strong consistency and strong consistency rate of the least squares estimator are obtained.     

A central limit theorem for the<Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> error of positive wavelet density estimator

The asymptotic distribution of the integrated squared error of positive wavelet density estimator is derived. It is shown that three different cases arise depending on the smoothness of the unknown density. In each case the asymptotic distribution is shown to be normal. A Martingale central limit theorem is used to prove the results.     

Constructive methods in convexC 2 interpolation using quartic splines

Using quartic splines on refined grids, we present a method for convexity preservingC ² interpolation which is successful for all strictly convex data sets. In the first stage, one suitable additional knot in each subinterval of the original data grid is fixed dependent on the given data values. In the second stage, a visually pleasant interpolant is selected by minimizing an appropriate choice functional.     

The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one,theta functions,trace formulas and the Selberg zeta function

We calculate the wave kernels for the classical rank-one symmetric spaces. The result is employed in order to provide a meromorphic extension of the theta function of an even-dimensional compact locally symmetric space of non-compact type. Moreover we give a short derivation of the Selberg trace formula. We discuss the relation between the right hand side of the functional equation of the Selberg zeta function, the Plancherel measure, Weyl's dimension formula and the wave kernel on the non-compact symmetric space and on its compact dual in an explicit manner.The first two authors were supported by the Sonderforschungsbereich 288 Differentialgeometrie und Quantenphysik founded by the Deutsche Forschungsgemeinschaft.     

Higher order asymptotics in estimation for two-sided Weibull type distributions

We consider the estimation problem of a location parameter on a sample of size n from a two-sided Weibull type density % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamOzaiaacIcacaWG4bGaeyOeI0IaeqiUdeNaaiykaiabg2da9iaa% doeacaGGOaGaeqySdeMaaiykaiGacwgacaGG4bGaaiiCaiaacIcacq% GHsislcaGG8bGaamiEaiabgkHiTiabeI7aXjaacYhadaahaaWcbeqa% aiabeg7aHbaakiaacMcaaaa!52AD!\[f(x - \theta ) = C(\alpha )\exp ( - |x - \theta |^\alpha )\] for –<x<, –<< and 1<a<3/2, where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Gaam4qaiaacIcacqaHXoqycaGGPaGaeyypa0JaeqySdeMaai4laiaa% cUhacaaIYaGaeu4KdCKaaiikaiaaigdacaGGVaGaeqySdeMaaiykai% aac2haaaa!4B0E!\[C(\alpha ) = \alpha /\{ 2\Gamma (1/\alpha )\} \]. Then the bound for the distribution of asymptotically median unbiased estimators is obtained up to the 2a-th order, i.e., the order % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamOBamaaCaaaleqabaGaeyOeI0IaaiikaiaaikdacqaHXoqycqGH% sislcaaIXaGaaiykaiaac+cacaaIYaaaaaaa!4444!\[n^{ - (2\alpha - 1)/2} \]. The asymptotic distribution of a maximum likelihood estimator (MLE) is also calculated up to the 2a-th order. It is shown that the MLE is not 2a-th order asymptotically efficient. The amount of the loss of asymptotic information of the MLE is given.     

On Adaptive Combination of Regression Estimators

Consider the problem of choosing between two estimators of the regression function, where one estimator is based on stronger assumptions than the other and thus the rates of convergence are different. We propose a linear combination of the estimators where the weights are estimated by Mallows' C _L . The adaptive estimator retains the optimal rates of convergence and is an extension of Stein-type estimators considered by Li and Hwang (1984, Ann. Statist., 12, 887-897) and related to an estimator in Burman and Chaudhuri (1999, Ann. Inst. Statist. Math. (to appear)).