ALL-BIAS DESIGNS FOR POLYNOMIAL SPLINE REGRESSION MODELS

Polynomial spline regression models of low degree have proved useful in modeling responses from designed experiments in science and engineering when simple polynomial models are inadequate. Where there is uncertainty in the number and location of the knots, or breakpoints, of the spline, then designs that minimize the systematic errors resulting from model misspecification may be appropriate. This paper gives a method for constructing such all‐bias designs for a single variable spline when the distinct knots in the assumed and true models come from some specified set. A class of designs is defined in terms of the inter‐knot intervals and sufficient conditions are obtained for a design within this class to be all‐bias under linear, quadratic and cubic spline models. An example of the construction of all‐bias designs is given.     

Adaptive tests of regression functions via multiscale generalized likelihood ratios

Many applications of nonparametric tests based on curve estimation involve selecting a smoothing parameter. The author proposes an adaptive test that combines several generalized likelihood ratio tests in order to get power performance nearly equal to whichever of the component tests is best. She derives the asymptotic joint distribution of the component tests and that of the proposed test under the null hypothesis. She also develops a simple method of selecting the smoothing parameters for the proposed test and presents two approximate methods for obtaining its P‐value. Finally, she evaluates the proposed test through simulations and illustrates its application to a set of real data.     

具扩充Jacobi结点的Lagrange插值逼近

研究了以扩充Jacobi多项式(1+x)Vn(x)的零点为基点的Lagrange插值多项式Ln(f,x)逼近/k)的一些问题.     

一阶非线性中立型方程解的渐近性与振动性

本文研究一类具有交系数和多偏差的一阶非线性中立型微分方程解的渐近性与振动性     

初等数学与高等数学的融合

本文从高等数学对初等数学的渗透，高观点下的初等数学，高等数学对初等数学的直接指导作用三个方面阐述了初等数学与高等数学的融合。     

一类多项式零点问题的证明

多项式零点问题的证明,已见于若干学报或教材．其方法可谓多种多样,但显复杂．本文先将Rolle定理加以推广,称之为“广义Rolle定理”,然后借助于该定理,证明一类著名的多项式零点值问题．     

Convergence rates for uniform confidence intervals based on local polynomial regression estimators

We investigate the convergence rates of uniform bias-corrected confidence intervals for a smooth curve using local polynomial regression for both the interior and boundary region. We discuss the cases when the degree of the polynomial is odd and even. The uniform confidence intervals are based on the volume-of-tube formula modified for biased estimators. We empirically show that the proposed uniform confidence intervals attain, at least approximately, nominal coverage. Finally, we investigate the performance of the volume-of-tube based confidence intervals for independent non-Gaussian errors.     

Exact distributions of two non-central generalized Wilks's statistics

We derive the exact expressions of the probability density function (pdf) and the cumulative distribution function (cdf) of Wilks's likelihood ratio criterion Λ and Wilks-Lawley's statistic U in the non-central linear and the non-central planar cases. Those expressions are given in rapidly converging infinite series and can be used for numerical computation. For applications, we compute the exact power of these statistics in a multivariate analysis of variance exercise, and show by simulation the precision of our analytic formulae.     

Cramér Asymptotic Efficiency in a Semiparametric Model

This article is concerned with efficient estimation in a semiparametric model. We consider pseudo maximum likelihood estimation and prove that the proposed estimator is asymptotically efficient in the sense of Cramér; that is, the estimator has the smallest mean squared error.     

Least squares estimation of varying‐coefficient hazard regression with application to breast cancer dose‐intensity data

To enhance modeling flexibility, the authors propose a nonparametric hazard regression model, for which the ordinary and weighted least squares estimation and inference procedures are studied. The proposed model does not assume any parametric specifications on the covariate effects, which is suitable for exploring the nonlinear interactions between covariates, time and some exposure variable. The authors propose the local ordinary and weighted least squares estimators for the varying‐coefficient functions and establish the corresponding asymptotic normality properties. Simulation studies are conducted to empirically examine the finite‐sample performance of the new methods, and a real data example from a recent breast cancer study is used as an illustration. The Canadian Journal of Statistics 37: 659–674; 2009 © 2009 Statistical Society of Canada