Solutions for a class of invex programming problems

A nonconvex constrained optimization problem is considered in which the constraints are of the form of generalized polynomials. An invexity kernel is established for this class of problem, and a consequent theorem gives sufficient conditions for the solutions of such problems.     

Root-n-consistent semiparametric estimation of partially linear models for weakly dependent observations

We consider estimation of the linear part in a partially linear model for absolutely regular observations. The estimator using random weights are proposed and the asymptotic normality of the estimator is established without compact support assumption.     

Exact D-optimal designs for a linear log contrast model with mixture experiment for three and four ingredients

This study investigates the exact D-optimal designs of the linear log contrast model using the mixture experiment suggested by Aitchison and Bacon-Shone (1984) and the design space restricted by Lim (1987) and Chan (1988). Results show that for three ingredients, there are six extreme points that can be divided into two non-intersect sets S₁ and S₂. An exact N-point D  -optimal design for N=3p+q,p≥1,1≤q≤2$N=3p+q,p\ge 1,1\le q\le 2$ arranges equal weight n/N,0≤n≤p$n/N,0\le n\le p$ at the points of S₁ (S₂) and puts the remaining weight (N−3n)/N$(N-3n)/N$ on the points of S₂ (S₁) as evenly as possible. For four ingredients and N=6p+q,p≥1,1≤q≤5$N=6p+q,p\ge 1,1\le q\le 5$, an exact N-point design that distributes the weights as evenly as possible among the six supports of the approximate D-optimal design is exact D-optimal.     

On the Uniform Strong Consistency of Local Polynomial Regression Under Dependence Conditions

Abstract

In this article, nonparametric estimators of the regression function, and its derivatives, obtained by means of weighted local polynomial fitting are studied. Consider the fixed regression model where the error random variables are coming from a stationary stochastic process satisfying a mixing condition. Uniform strong consistency, along with rates, are established for these estimators. Furthermore, when the errors follow an AR(1) correlation structure, strong consistency properties are also derived for a modified version of the local polynomial estimators proposed by Vilar-Fernández and Francisco-Fernández (Vilar-Fernández, J. M., Francisco-Fernández, M. (2002 Vilar-Fernández, J. M. and Francisco-Fernández, M. 2002. Local polynomial regression smoothers with AR-error structure. TEST, 11(2): 439–464.  [Google Scholar]). Local polynomial regression smoothers with AR-error structure. TEST 11(2):439–464).     

The Estimation and Testing of the Cointegration Order Based on the Frequency Domain

ABSTRACT

This article proposes a method to estimate the degree of cointegration in bivariate series and suggests a test statistic for testing noncointegration based on the determinant of the spectral density matrix for the frequencies close to zero. In the study, series are assumed to be I(d), 0 < d ? 1, with parameter d supposed to be known. In this context, the order of integration of the error series is I(d ? b), b ∈ [0, d]. Besides, the determinant of the spectral density matrix for the dth difference series is a power function of b. The proposed estimator for b is obtained here performing a regression of logged determinant on a set of logged Fourier frequencies. Under the null hypothesis of noncointegration, the expressions for the bias and variance of the estimator were derived and its consistency property was also obtained. The asymptotic normality of the estimator, under Gaussian and non-Gaussian innovations, was also established. A Monte Carlo study was performed and showed that the suggested test possesses correct size and good power for moderate sample sizes, when compared with other proposals in the literature. An advantage of the method proposed here, over the standard methods, is that it allows to know the order of integration of the error series without estimating a regression equation. An application was conducted to exemplify the method in a real context.     

Finite Sample Comparison of Parametric,Semiparametric, and Wavelet Estimators of Fractional Integration

ABSTRACT

In this paper we compare through Monte Carlo simulations the finite sample properties of estimators of the fractional differencing parameter, d. This involves frequency domain, time domain, and wavelet based approaches, and we consider both parametric and semiparametric estimation methods. The estimators are briefly introduced and compared, and the criteria adopted for measuring finite sample performance are bias and root mean squared error. Most importantly, the simulations reveal that (1) the frequency domain maximum likelihood procedure is superior to the time domain parametric methods, (2) all the estimators are fairly robust to conditionally heteroscedastic errors, (3) the local polynomial Whittle and bias-reduced log-periodogram regression estimators are shown to be more robust to short-run dynamics than other semiparametric (frequency domain and wavelet) estimators and in some cases even outperform the time domain parametric methods, and (4) without sufficient trimming of scales the wavelet-based estimators are heavily biased.     

Psychotropic drug classification based on sleep–wake behaviour of rats

Summary.  The aim of the paper is to present methodology for the classification of potential psychotropic drugs on the basis of their activity. We first sketch the background of this class of drugs and then zoom in on so-called pharmacoelectroencephalogram studies. These data pose some statistical challenges. For classification purposes, we propose a flexible hierarchical discriminant analysis tool, allowing us to take the specific nature of the drug class into account, as well as the features of the mixed models, in combination with fractional polynomials, fitted to the electroencephalogram data. The method is evaluated against the background of existing methods. The method's performance is studied by using a comprehensive analysis of a large electroencephalogram data set.     

一道IMO试题的推广及证明

将 4 2届国际数学奥林匹克竞赛 (IMO)中的一个不等式问题加以推广 ,并给出其证明     

Maximum variance of order statistics from symmetric populations revisited

We consider i.i.d. samples of size n with symmetric non-degenerate parent distributions and finite variances. Papadatos [A note on maximum variance of order statistics from symmetric populations, Ann. Inst. Statist. Math. 48 (1997), pp. 117–121] proved that the maximal variance of each non-extreme order statistic, expressed in the population variance units, is attained in a one-parametric family of symmetric two- and three-point distributions. The parameters of the extreme variance distributions coincide with the arguments maximizing some polynomials of degree 2n?1 over a finite interval. The bounds for variances are equal to the maximal values of the polynomials. We present a more precise solution to the problem by applying the variation diminishing property of Bernstein polynomials.