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Density Approximation by Summary Statistics: An Information-Theoretic Approach
Zvi Gilula and Shelby J. Haberman
Scandinavian Journal of Statistics
Vol. 27, No. 3 (Sep., 2000), pp. 521-534
Published by: Wiley on behalf of Board of the Foundation of the Scandinavian Journal of Statistics
Stable URL: http://www.jstor.org/stable/4616620
Page Count: 14
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In the case of exponential families, it is a straightforward matter to approximate a density function by use of summary statistics; however, an appropriate approach to such approximation is far less clear when an exponential family is not assumed. In this paper, a maximin argument based on information theory is used to derive a new approach to density approximation from summary statistics which is not restricted by the assumption of validity of an underlying exponential family. Information-theoretic criteria are developed to assess loss of predictive power of summary statistics under such minimal knowledge. Under these criteria, optimal density approximations in the maximin sense are obtained and shown to be related to exponential families. Conditions for existence of optimal density approximations are developed. Applications of the proposed approach are illustrated, and methods for estimation of densities are provided in the case of simple random sampling. Large-sample theory for estimates is developed.
Scandinavian Journal of Statistics © 2000 Board of the Foundation of the Scandinavian Journal of Statistics