TY - JOUR
T1 - Density approximation by summary statistics
T2 - An information-theoretic approach
AU - Gilula, Zvi
AU - Haberman, Shelby J.
PY - 2000
Y1 - 2000
N2 - 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.
AB - 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.
KW - Exponential families
KW - Optimal predictive densities
KW - Probability prediction
UR - http://www.scopus.com/inward/record.url?scp=0034258303&partnerID=8YFLogxK
U2 - 10.1111/1467-9469.00204
DO - 10.1111/1467-9469.00204
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AN - SCOPUS:0034258303
SN - 0303-6898
VL - 27
SP - 521
EP - 534
JO - Scandinavian Journal of Statistics
JF - Scandinavian Journal of Statistics
IS - 3
ER -