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标题：Approximation of the expected value of the harmonic mean and some applications
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作者：Calyampudi Radhakrishna Rao ; Xiaoping Shi ; Yuehua Wu 等
期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2014
卷号：111
期号：44
页码：15681-15686
DOI：10.1073/pnas.1412216111
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：SignificanceThe harmonic mean (HM) filter is better at removing positive outliers than the arithmetic mean (AM) filter. There are especially difficult issues when an accurate evaluation of expected HM is needed such as, for example, in image denoising and marginal likelihood evaluation. A major challenge is to develop a higher-order approximation of the expected HM when the central limit theorem is not applicable. A two-term approximation of the expected HM is derived in this paper. This approximation enables us to develop a new filtering procedure to denoise the noisy image with an improved performance, and construct a truncated HM estimator with a faster convergence rate in marginal likelihood evaluation. Although the harmonic mean (HM) is mentioned in textbooks along with the arithmetic mean (AM) and the geometric mean (GM) as three possible ways of summarizing the information in a set of observations, its appropriateness in some statistical applications is not mentioned in textbooks. During the last 10 y a number of papers were published giving some statistical applications where HM is appropriate and provides a better performance than AM. In the present paper some additional applications of HM are considered. The key result is to find a good approximation to [IMG]f1.gif" ALT="Formula" BORDER="0">, the expectation of the harmonic mean of n observations from a probability distribution. In this paper a second-order approximation to [IMG]f1.gif" ALT="Formula" BORDER="0"> is derived and applied to a number of problems.
关键词：harmonic mean ; second-order approximation ; arithmetic mean ; image denoising ; marginal likelihood