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https://hdl.handle.net/2142/19231
Description
Title
Some topics in sequential density estimation
Author(s)
Kundu, Subrata
Issue Date
1994
Doctoral Committee Chair(s)
Martinsek, Adam T.
Department of Study
Statistics
Discipline
Statistics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Statistics
Language
eng
Abstract
Let $X\sb1, X\sb2, \... X\sb{n}$ be i.i.d random variables with common unknown density function f. Here we are interested in estimating the unknown density f with bounded Mean Integrated Absolute Error (MIAE). Devroye and Gyorfi (1985) obtained asymptotic bounds for the MIAE in estimating f by a kernel estimate $\ f\sb{n}.$ Using these bounds one can identify an appropriate sample size such that the MIAE is smaller than some pre-assigned quantity w $>$ 0. Hence there is no fixed sample size that can be used to solve the problem of bounding the MIAE. In this work we propose stopping rules and two-stage procedures for bounding the $L\sb1$ distance. We show that these procedures are asymptotically optimal in a certain sense as w $\to$ 0.
The choice of the bandwidth plays a key role in the performance of the estimators. The optimal bandwidth depends on the unknown density f and one relies on the data driven choices of bandwidth for improving the performance. A two stage procedure involving data dependent bandwidth selection is proposed. Optimality of this two stage procedure is established.
The second part of the thesis addresses the problem of estimating the unknown density with bounded $L\sb{p}$ error for some p $>$ 2. Asymptotic bounds for the $L\sb{p}$ distance obtained by Bretagnolle and Huber (1989) are used to identify an appropriate non-random sample size such that the $L\sb{p}$ distance between the true density and the estimated density is smaller than some pre-assigned quantity w $>$ 0. It is observed that no fixed sample size can be used to solve this problem, since the optimal sample size depends on unknown f. Here also we propose two-stage and sequential procedures for bounding the $L\sb{p}$ distance. These procedures are shown to be optimal.
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