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Density Approximation by Summary Statistics: An Information‐theoretic Approach

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  • Zvi Gilula
  • S. J. Haberman

Abstract

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.

Suggested Citation

  • Zvi Gilula & S. J. Haberman, 2000. "Density Approximation by Summary Statistics: An Information‐theoretic Approach," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(3), pages 521-534, September.
    • Handle: RePEc:bla:scjsta:v:27:y:2000:i:3:p:521-534
    DOI: 10.1111/1467-9469.00204
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    Cited by:

    1. Shelby J. Haberman, 2015. "Pseudo-Equivalent Groups and Linking," Journal of Educational and Behavioral Statistics, , vol. 40(3), pages 254-273, June.
    2. Emanuele Taufer & Sudip Bose & Aldo Tagliani, 2009. "Optimal predictive densities and fractional moments," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(1), pages 57-71, January.

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