This thesis represents an attempt to generalize the classical Theory of Optimal Design to popular regression models, based on Rational and Spline approximations. The problem of finding optimal designs for such models can be reduced to solving certain minimax problems. Explicit solutions to such
problems can be obtained only in a few selected models, such as polynomial regression.
Even when an optimal design can be found, it has, from the point of view of modern nonparametric regression, certain drawbacks. For example, in the polynomial regression case, the optimal design crucially depends on the degree m of approximating polynomial.
Hence, it can be used only when such degree is given/known in advance.
We present a partial, but practical, solution to this problem. Namely, the so-called Super Chebyshev Design has been found, which does not depend on the degree m of the underlying
polynomial regression in a large range of m, and at the same time is asymptotically more than 90% efficient. Similar results are obtained in the case of rational regression, even though the exact form of optimal design in this case remains unknown.
Optimal Designs in the case of Spline Interpolation are also currently unknown. This problem, however, has a simple solution in the case of Cardinal Spline Interpolation. Until recently, this model has been practically unknown in modern nonparametric
regression. We demonstrate the usefulness of Cardinal Kernel Spline Estimates in nonparametric regression, by proving their
asymptotic optimality, in certain classes of smooth functions. In this way, we have found, for the first time, a theoretical justification of a well known empirical observation, by which cubic splines suffice, in most practical applications. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2007-07-18 16:06:06.767
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OKQ.1974/448 |
| Date | 19 July 2007 |
| Creators | Cho, Jaerin |
| Contributors | "Queen's University (Kingston, Ont.). Theses (Queen's University (Kingston, Ont.))" |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Format | 677946 bytes, application/pdf |
| Rights | "This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner." |
| Relation | "Canadian theses" |
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