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．     

Mini-max-risk and mini-mean-risk inferences for a partially piecewise regression

We consider a partially piecewise regression in which the main regression coefficients are constant in all subdomains, but the extraessential regression function is variable in different pieces and is difficult to be estimated. Under this situation, two new regression methodologies are proposed under the criteria of mini-max-risk and mini-mean-risk. The resulting models can describe the regression relations in maximum-risk and mean-risk environments, respectively. A two-stage estimation procedure, together with a composite method, is introduced. The asymptotic normality of the estimators is established, the standard convergence rate and efficiency are achieved. Some unusual features of the new estimators and predictions, and the related variable selection are discussed for a comprehensive comparison. Simulation studies and a real-financial example are given to illustrate the new methodologies.     

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.     

Trajectory of problem behaviours of Korean adopted children: using piecewise hierarchical linear growth modelling

This study examined the trajectory of problem behaviours in domestically adopted children in South Korea as they aged. This study used the Panel Study on Korean Adopted Children's longitudinal, three‐wave data (2006, 2008 and 2010). Although data were collected at three time points, our data consisted of six time points, which covered the 5–10 years age range of the adopted children at the time of survey. One hundred sixty‐four children were included in the analysis, 75 of whom contributed to one time point, 74 to two time points and 15 to three time points. The trajectory of the problem behaviours of adopted children was examined using a piecewise hierarchical linear growth model. Because the initial exploration of the data suggested non‐linear changes in behaviour problems over time, we split the growth trajectory into two time periods: Time 1 (5–7 years) and Time 2 (7–10 years). A two‐rate model was used to estimate separate slopes for the two time periods. Results showed that the externalizing and internalizing problems of adopted children have different trajectories. Internalizing problems did not show significant changes after 5 years of age, while externalizing problems increased until 7 years of age and decreased significantly thereafter.