Orthogonal projections and the geometry of estimating functions |
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| Institution: | 1. Departments of Internal Medicine, Kyungpook National University School of Medicine, Daegu, South Korea;2. Biochemistry and Cell Biology, Kyungpook National University School of Medicine, Daegu, South Korea;3. Cell and Matrix Research Institute, Kyungpook National University School of Medicine, Daegu, South Korea;4. Preventive Medicine, Kyungpook National University School of Medicine, Daegu, South Korea;5. Thoracic Surgery, Kyungpook National University School of Medicine, Daegu, South Korea;6. Center for Lung Cancer, National Cancer Center, Goyang, South Korea;7. Lung and Esophageal Cancer Clinic, Chonnam National University Hwasun Hospital, Chonnam, South Korea;8. Department of Thoracic and Cardiovascular Surgery, Seoul National University School of Medicine, Seoul, South Korea;9. Department of Internal Medicine, Catholic University of Daegu, School of Medicine, Daegu, South Korea;10. Department of Internal Medicine, Pusan National University School of Medicine, Busan, South Korea |
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| Abstract: | In this paper, a notion of generalized inner product spaces is introduced to study optimal estimating functions. The basic technique involves an idea of orthogonal projection first introduced by Small and McLeish (1988, 1989, 1991, 1992, 1994). A characterization of orthogonal projections in generalized inner product spaces is given. It is shown that the orthogonal projection of the score function into a linear subspace of estimating functions is optimal in that subspace, and a characterization of optimal estimating functions is given. As special cases of the main results of this paper, we derive the results of Godambe (1985) on the foundation of estimation in stochastic processes, the result of Godambe and Thompson (1989) on the extension of quasi-likelihood, and the generalized estimating equations for multivariate data due to Liang and Zeger (1986). Also we have derived optimal estimating functions in the Bayesian framework. |
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