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A SPECTRAL METHOD FOR DECONVOLVING A DENSITY

Published online by Cambridge University Press:  11 October 2010

Marine Carrasco  and
Jean-Pierre Florens
Marine Carrasco*
Affiliation:
University of Montreal
Jean-Pierre Florens
Affiliation:
Toulouse School of Economics
*
*Address correspondence to Marine Carrasco, Universite de Montreal, Departement de Sciences Economiques, CP 6128, succ Centre Ville, Montreal, QC H3C3J7, Canada; e-mail: marine.carrasco@umontreal.ca.
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Abstract

We propose a new estimator for the density of a random variable observed with an additive measurement error. This estimator is based on the spectral decomposition of the convolution operator, which is compact for an appropriate choice of reference spaces. The density is approximated by a sequence of orthonormal eigenfunctions of the convolution operator. The resulting estimator is shown to be consistent and asymptotically normal. While most estimation methods assume that the characteristic function (CF) of the error does not vanish, we relax this assumption and allow for isolated zeros. For instance, the CF of the uniform and symmetrically truncated normal distributions have isolated zeros. We show that, in the presence of zeros, the density is identified even though the convolution operator is not one-to-one. We propose two consistent estimators of the density. We apply our method to the estimation of the measurement error density of hourly income collected from survey data.

Type
ARTICLES
Information
Econometric Theory , Volume 27 , Issue 3: SPECIAL ISSUE ON INVERSE PROBLEMS IN ECONOMETRICS , June 2011 , pp. 546 - 581
Copyright
Copyright © Cambridge University Press 2011

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