Nonparametric Density Estimation under Unimodality and Monotonicity Constraints

We introduce a recursive method for estimating a probability density subject to constraints of unimodality or monotonicity. It uses an empirical estimate of the probability transform to construct a sequence of maps of a known template, which satisfies the constraints. The algorithm may be employed w...

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Bibliographic Details
Published inJournal of computational and graphical statistics Vol. 8; no. 1; pp. 1 - 21
Main Authors Cheng, Ming-Yen, Gasser, Theo, Hall, Peter
Format Journal Article
LanguageEnglish
Published Taylor & Francis Group 01.03.1999
American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America
Subjects
Curve estimation
Data smoothing
Density
Density estimation
Estimation bias
Estimation methods
Estimators
Interval estimators
Isotonic regression
Iteration
Kernel methods
Mode
Mode testing
Probability transform
Recursion
Sample size
Smoothing
Statistical variance
Turning point
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Summary:We introduce a recursive method for estimating a probability density subject to constraints of unimodality or monotonicity. It uses an empirical estimate of the probability transform to construct a sequence of maps of a known template, which satisfies the constraints. The algorithm may be employed without a smoothing step, in which case it produces step-function approximations to the sampling density. More satisfactorily, a certain amount of smoothing may be interleaved between each recursion, in which case the estimate is smooth. The amount of smoothing may be chosen using a standard cross-validation algorithm. Unlike other methods for density estimation, however, the recursive approach is robust against variation of the amount of smoothing, and so choice of bandwidth is not critical.
ISSN:1061-8600
1537-2715
DOI:10.1080/10618600.1999.10474798