THEORY OF CANONICAL MOMENTS AND ITS APPLICATIONS IN POLYNOMIAL REGRESSION

Consider a regression model Y(x) = (beta)(,0) + (beta)(,1)x +...+ (beta)(,m)x(\u27m) + (epsilon) on an interval {a,b}, where (epsilon) (TURN) N(0,(sigma)(\u272)). Suppose the least squares method is used to estimate some linear combinations of the (beta)\u27s. The optimal design theory concerns the choice of the allocation of the observations to accomplish the estimation in an optimal way. This amounts to dealing with the minimization of some functionals of the covariance matrix. The present work uses canonical moments as a general tool to solve optimal design problems. This approach not only unifies many old results in a simpler way, but also provides various new optimal designs relating to: weighted D-optimal design, weighted D(,s)-optimal design, trigonometric regression, rotation design, weighted extrapolation design and integrated variance design, etc. One of the drawbacks of the classical optimal design theory is that it assumes the experimenter knows the model exactly. To guard against the possible model violations, we seek robust designs via Stigler\u27s approach (Stigler 1971, JASA). The designs found by the method of the canonical moments, turn out to have high efficiency in estimating the regression function and have reasonable power to check the model. The method of canonical moments is also used to study the design for comparison of models--an important topic in linear model theory. Relations between canonical moments and moments, orthogonal polynomials and measures are also discussed

The reliability of exchangeable binary systems

Assuming that the components of a system are Bernoulli and positive dependent by mixture, we can estimate the reliability of a k-out-of-n:F system, a consecutive k-out-of-n:F system and a circular consecutive k-out-of-n:F system by using canonical moments.k-out-of-n:F system consecutive k-out-of-n:F system circular consecutive k-out-of-n:F system canonical moments mixture reliability positive dependent

Empirical Likelihood for Partially Linear Models

In this paper, we consider the application of the empirical likelihood method to partially linear model. Unlike the usual cases, we first propose an approximation to the residual of the model to deal with the nonparametric part so that Owen's (1990) empirical likelihood approach can be applied. Then, under quite general conditions, we prove that the empirical log-likelihood ratio statistic is asymptotically chi-squared distributed. Therefore, the empirical likelihood confidence regions can be constructed accordingly.partially linear model, empirical likelihood, nonparametric likelihood ratio, sieve approximation, weight functions

A note on D-optimal designs for models with and without an intercept

Optimal design, Polynomial regression, Product model, Weighing design,

Study on ultra-precision compliant mechanisms for nanotechnology applications

Including: 2 parts. Compliant mechanisms provide motion through elastic deformation under the action of external loads. These mechanisms are key functional members in many today's precision machines and devices, such as precision micro-positioning stages, micro actuators, microelectromechanical systems (MEMS) and robots, where micron or even nanometric resolution and accuracy are required for the motion. On the contrary to rigid-body mechanisms, compliant mechanisms consist of monolithic construction without rigid joints or sliders. Thus, they effectively eliminate the wear, backlash, lubrication, and friction problems, which are often encountered by rigid-body mechanisms

Study on ultra-precision compliant mechanisms for nanotechnology applications

Including: 2 parts. Compliant mechanisms provide motion through elastic deformation under the action of external loads. These mechanisms are key functional members in many today's precision machines and devices, such as precision micro-positioning stages, micro actuators, microelectromechanical systems (MEMS) and robots, where micron or even nanometric resolution and accuracy are required for the motion. On the contrary to rigid-body mechanisms, compliant mechanisms consist of monolithic construction without rigid joints or sliders. Thus, they effectively eliminate the wear, backlash, lubrication, and friction problems, which are often encountered by rigid-body mechanisms. Furthermore, their monolithic construction makes the costly assembly process unnecessary and the integration of smart sensors and actuators possible