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.     

LOG-PERIODOGRAM ESTIMATION OF LONG MEMORY VOLATILITY DEPENDENCIES WITH CONDITIONALLY HEAVY TAILED RETURNS

Many recent papers have used semiparametric methods, especially the log-periodogram regression, to detect and estimate long memory in the volatility of asset returns. In these papers, the volatility is proxied by measures such as squared, log-squared, and absolute returns. While the evidence for the existence of long memory is strong using any of these measures, the actual long memory parameter estimates can be sensitive to which measure is used. In Monte-Carlo simulations, I find that if the data is conditionally leptokurtic, the log-periodogram regression estimator using squared returns has a large downward bias, which is avoided by using other volatility measures. In United States stock return data, I find that squared returns give much lower estimates of the long memory parameter than the alternative volatility measures, which is consistent with the simulation results. I conclude that researchers should avoid using the squared returns in the semiparametric estimation of long memory volatility dependencies.     

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。     

Finite Sample Performance in Cointegration Analysis of Nonlinear Time Series with Long Memory

Nonlinear functions of multivariate financial time series can exhibit long memory and fractional cointegration. However, tools for analysing these phenomena have principally been justified under assumptions that are invalid in this setting. Determination of asymptotic theory under more plausible assumptions can be complicated and lengthy. We discuss these issues and present a Monte Carlo study, showing that asymptotic theory should not necessarily be expected to provide a good approximation to finite-sample behavior.     

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.