On the Edgeworth expansion and the <Emphasis Type="Italic">M</Emphasis> out of <Emphasis Type="Italic">N</Emphasis> bootstrap accuracy for a Studentized trimmed mean

We investigate the second order accuracy of the M out of N bootstrap for a Studentized trimmed mean using the Edgeworth expansion derived in a previous paper. Some simulations, which support our theoretical results, are also given. The effect of extrapolation in conjunction with the M out of N bootstrap for Studentized trimmed means is briefly discussed. As an auxiliary result we obtain a Bahadur’s type representation for an M out of N bootstrap quantile. Our results supplement previous work on (Studentized) trimmed means by Hall and Padmanabhan [13], Bickel and Sakov [7], and Gribkova and Helmers [11].      

A functional analysis of NOx levels: location and scale estimation and outlier detection

This paper analyzes the NOx levels measured by a control station near a power plant by using techniques for functional data. First, we test for differences between the levels on working and non working days. Second, we obtain several location estimators and confidence sets of the center of the functional distribution. Third, we provide scale estimators and confidence sets of the dispersion of the functional distribution. Finally, a distance based procedure provides a criterion to determinate the presence of outlying observations, which allows to detect relevant NOx levels.     

An efficient shrinkage bootstrap bias estimator for smooth functions of sample means

Summary  Since it is not always possible to calculate bootstrap estimators, they are usually approximated by simulation. In this article, we propose a bootstrap bias estimator for smooth functions of sample means that has less mean squared error, due to the simulation process, than the ordinary bootstrap. The estimator is based on shrinking the bootstrap mean towards the original sample mean. It can easily be implemented while demanding almost no additional computational effort.     

A LIL and limit distributions for trimmed sums of random vectors attracted to operator semi-stable laws

Let θ∈ Rdbe a unit vector and let X,X1,X2,...be a sequence of i.i.d.Rd-valued random vectors attracted to operator semi-stable laws.For each integer n ≥ 1,let X1,n ≤···≤ Xn,n denote the order statistics of X1,X2,...,Xn according to priority of index,namely | X1,n,θ | ≥···≥ | Xn,n,θ |,where ·,· is an inner product on Rd.For all integers r ≥ 0,define by(r)Sn = n-ri=1Xi,n the trimmed sum.In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums(r)Sn.Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded.A stochastically compactness of(r)Sn is obtained.     

A variant of Jensen-Steffensen's inequality and quasi-arithmetic means

A variant of Jensen-Steffensen's inequality is proved. Necessary and sufficient conditions for the equality in Jensen-Steffensen's inequality are established. Several inequalities involving more than two monotonic functions and generalized quasi-arithmetic means with not only positive weights are proved. It is shown that such generalized quasi-arithmetic means have the same comparability properties as those with positive weights.     

Comparison of means generated by two functions and a measure

Given two continuous functions f,g:I→R such that g is positive and f/g is strictly monotone, and a probability measure μ on the Borel subsets of [0,1], the two variable mean Mf,g;μ:I²→I is defined by

     

Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve F(x,y)=0 in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces p(s,t) and q(u,v) defined on (s,t)∈[0,1]² and (u,v)∈[0,1]². The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given. Mathematics subject classification (2000)  65D17     

加权幂平均不等式的加强

对加权幂平均不等式的加强作了全面的研究,推广了一个加强的加权算术-几何不等式．     

Decrease rate of the probabilities ofε-deviations for the means of stationary processes

The asymptotic behavior asn → ∞ of the normed sumsσn =n ⁻¹ Σ _k ⁼⁰ⁿ⁻¹ Xk for a stationary processX = (X _n ,n ∈ ℤ) is studied. For a fixedε > 0, upper estimates for P(sup _k≥n |σ _k | ≥ε) asn → ∞ are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 366–372, September, 1998.     

Approximation to the distribution of the least squares estimators in two dimensional cosine models by randomly weighted bootstrap

Recently, Kundu and Gupta （Metrika, 48：83 C 97, 1998） established the asymptotic normality of the least squares estimators in the two dimensional cosine model. In this paper, we give the approximation to the general least squares estimators by using random weights which is called the Bayesian bootstrap or the random weighting method by Rubin （Annals of Statistics, 9：130 C 134, 1981） and Zheng （Acta Math. Appl. Sinica （in Chinese）, 10（2）： 247 C 253, 1987）. A simulation study shows that this approximation works very well.     

向径函数的球面平均及其点态收敛性

设球面平均函数为Mt（f）（x）=∫s^n-1f（x-ty′）dσ（y′）,则当f∈L^p（R^n）是向径函数，n≥3,1〈p≤n/（n-1）时，lim（t→0）Mt（f）（x）=f（x）几乎处处成立。     

Space-time means and solutions to a class of nonlinear parabolic equations

Cauchy problem and initial boundary value problem for nonlinear parabolic equation inCB([0,T):L ^p ) orL ^q (0,T; L ^p ) type space are considered. Similar to wave equation and dispersive wave equation, the space-time means for linear parabolic equation are shown and a series of nonlinear estimates for some nonlinear functions are obtained by space-time means. By Banach fixed point principle and usual iterative technique a local mild solution of Cauchy problem or IBV problem is constructed for a class of nonlinear parabolic equations inCB([0,T);L ^p orL ^q (0,T; L ^p ) with ϕ(x)∈L ^r . In critical nonlinear case it is also proved thatT can be taken as infinity provided that ||ϕ(x)||_r is sufficiently small, where (p,q,r) is an admissible triple. Project supported by the National Natural Science Foundation of China (Grant No. 19601005).     

中国股市的羊群效应的ARCH检验模型与实证分析

本文提出一种检验羊群效应的方法 ,即通过检验个股截面收益的绝对偏差 (CSA D)与市场收益的非线性关系 ,来判断羊群效应是否显著 ,并对我国沪深两市的羊群效应进行了实证分析 ,结果发现我国沪深两市存在一定程度的羊群效应 .     

Local monotonicity properties of two-variable Gini means and the comparison theorem revisited

In this paper we examine local monotonicity properties of the so-called Gini means, defined by

     

On the Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system

The main aim of this paper is to prove that the maximal operator σ ₀ ^k*:= sup _n ∣σ _n,n ^k ∣ of the Fejér means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space H _1/2 to the space weak-L _1/2. Supported by the Hungarian NFSR (OTKA), grant no. M 36511/2001, T 048780.     

Mixed means over balls and annuli and lower bounds for operator norms of maximal functions

In this paper we prove mixed-means inequalities for integral power means of an arbitrary real order, where one of the means is taken over the ball , centered at and of radius , δ>0. Therefrom we deduce the corresponding Hardy-type inequality, that is, the operator norm of the operator S_δ which averages over , introduced by Christ and Grafakos in Proc. Amer. Math. Soc. 123 (1995) 1687-1693. We also obtain the operator norm of the related limiting geometric mean operator, that is, Carleman or Levin-Cochran-Lee-type inequality. Moreover, we indicate analogous results for annuli and discuss estimations related to the Hardy-Littlewood and spherical maximal functions.     

平面上有限个点到直线的距离和最小的问题

讨论并解决了平面上有限个点(x1,y1),(x2,y2),…,(xn,yn)(1)到直线的距离和d=∑ni=1|xi-c|(当直线为x=c时),∑ni=1|yi-axi-b|1+a2(当直线为y=ax+b时)最小的问题,它源于全最小一乘法.     

Another approach to asymptotic expansions for Euler’s approximations of semigroups

In [2], optimal bounds for the remainder terms in asymptotic expansions for Euler’s approximations of semigroups were derived. The approach was based on applications of the Fourier-Laplace transforms, which allowed one to reduce the problem to estimation of error terms in the Law of Large Numbers. In this paper, we propose an alternative (direct) approach based on application of certain integro-differential identities (the so-called multiplicative representations of differences). Such identities were introduced by Bentkus [3] and applied (see Bentkus and Paulauskas [4]) to derive the optimal convergence rates in Chernoff-type lemmas and Euler’s approximations of semigroups. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 267–284, April–June, 2006.     

On Mad and Comedians

A popular robust measure of dispersion of a random variable (rv) X is the median absolute deviation from the median med(|X - med(X)|), MAD for short, which is based on the median med(X) of X. By choosing Y = X, the MAD turns out to be a special case of the comedian med((X - med(X))(Y - med(Y))), which is a robust measure of covariance between rvs X and Y. We investigate the comedian in detail, in particular in the normal case, and establish strong consistency and asymptotic normality of empirical counterparts. This leads to a robust competitor of the coefficient of correlation as an asymptotic level--statistic for testing independence of X and Y. An example shows the weird fact that knowledge of the population med(X) does not necessarily pay (in the sense of asymptotic relative efficiency) when estimating the MAD.