THEORY OF CANONICAL MOMENTS AND ITS APPLICATIONS IN POLYNOMIAL REGRESSION

TAI SHING LAU, Purdue University

Abstract

Consider a regression model Y(x) = (beta)(,0) + (beta)(,1)x +...+ (beta)(,m)x('m) + (epsilon) on an interval {a,b}, where (epsilon) (TURN) N(0,(sigma)('2)). Suppose the least squares method is used to estimate some linear combinations of the (beta)'s. 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's 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.

Degree

Ph.D.

Subject Area

Statistics

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