The Annals of Statistics

Nonparametric Density Estimation with a Parametric Start

Nils Lid Hjort and Ingrid K. Glad

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The traditional kernel density estimator of an unknown density is by construction completely nonparametric in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example, the normal, while not losing much in precision when the true density is far from the parametric class. The idea is to multiply an initial parametric density estimate with a kernel-type estimate of the necessary correction factor. This works well in cases where the correction factor function is less rough than the original density itself. Extensive comparisons with the kernel estimator are carried out, including exact analysis for the class of all normal mixtures. The new method, with a normal start, wins quite often, even in many cases where the true density is far from normal. Procedures for choosing the smoothing parameter of the estimator are also discussed. The new estimator should be particularly useful in higher dimensions, where the usual nonparametric methods have problems. The idea is also spelled out for nonparametric regression.

Article information

Ann. Statist., Volume 23, Number 3 (1995), 882-904.

First available in Project Euclid: 11 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Bandwidth selection correction factor kernel methods lowering the bias semiparametric density estimation test cases


Hjort, Nils Lid; Glad, Ingrid K. Nonparametric Density Estimation with a Parametric Start. Ann. Statist. 23 (1995), no. 3, 882--904. doi:10.1214/aos/1176324627.

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