2026-09-18T19:37:38Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/19361 2023-07-10 col_2142_5131 col_2142_17362 com_2142_5130 com_2142_17361 com_2142_8903

Barron, Andrew Clarke, Bertrand Salem 2011-05-07T12:05:09Z 2011-05-07T12:05:09Z 10000-01-01 1989 In many areas of application of statistics one has a relevent parametric family of densities and wishes to estimate the density from a random sample. In such cases one can use the family to generate an estimator. We fix a prior and consider the properties of the predictive density as an estimator of the density. We examine the cumulative risk of the estimator and its cumulative Bayes risk under Kullback-Leibler loss. Those two mathematical quantities appear in other contexts with different interpretations. Aside from density estimation, the first quantity occurs in source coding, and hypothesis testing, and the second occurs in source coding, channel coding, and asymptotic convergence of the posterior to a normal. In the first chapter we state our two main results, give some examples, and discuss the applications of our results to those areas. Our two key results amount to two senses in which the Kullback-Leibler distance between the n-fold product of a distribution in a parametric family and a mixture of such distributions over the parametric family increases as the logarithm of the sample size, provided that in the mixing some mass is assigned near the true distribution. The first is a direct examination of the Kullback-Leibler distance, the second is an examination of the Kullback-Leibler distance after it has again been averaged with respect to the prior. We prove that each is of the form one half the dimension of the parameter times the logarithm of the sample size plus a constant. In both cases the constant is identified. The key technique for the first result is Laplace integration which gives upper and lower bounds which are asymptotically tight and can be made uniformly good over compact sets in the parameter space. When the parameter space is no longer compact it becomes advantageous to use different techniques in obtaining upper and lower bounds. The convergence holds in an average sense not pointwise uniform. For the upper bound we use an inequality due to Barron (1988) so as to set up an application of the dominated convergence theorem, and for a lower bound we use the fact that the normal has maximal entropy for constrained variance. Made available in DSpace on 2011-05-07T12:05:09Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9010836.pdf: 3628032 bytes, checksum: 78de6fae154ec2c807c1f6f4d9cb04f0 (MD5) Previous issue date: 1989 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:36:26Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:14:41-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9010836 (UMI)AAI9010836 http://hdl.handle.net/2142/19361 eng Copyright 1989 Clarke, Bertrand Salem Statistics Asymptotic cumulative risk and Bayes risk under entropy loss, with applications text Statistics Statistics University of Illinois at Urbana-Champaign Dissertation Ph.D.