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INFERENCE ON A DISTRIBUTION FROM NOISY DRAWS

Published online by Cambridge University Press:  18 August 2022

Koen Jochmans  and
Martin Weidner
Koen Jochmans*
Affiliation:
Université Toulouse 1 Capitole
Martin Weidner
Affiliation:
University of Oxford
*
Address correspondence to Koen Jochmans, Toulouse School of Economics, Université Toulouse 1 Capitole, 1 esplanade de l’Université, 31080 Toulouse, France; e-mail: koen.jochmans@tse-fr.eu.
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Abstract

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We consider a situation where the distribution of a random variable is being estimated by the empirical distribution of noisy measurements of that variable. This is common practice in, for example, teacher value-added models and other fixed-effect models for panel data. We use an asymptotic embedding where the noise shrinks with the sample size to calculate the leading bias in the empirical distribution arising from the presence of noise. The leading bias in the empirical quantile function is equally obtained. These calculations are new in the literature, where only results on smooth functionals such as the mean and variance have been derived. We provide both analytical and jackknife corrections that recenter the limit distribution and yield confidence intervals with correct coverage in large samples. Our approach can be connected to corrections for selection bias and shrinkage estimation and is to be contrasted with deconvolution. Simulation results confirm the much-improved sampling behavior of the corrected estimators. An empirical illustration on heterogeneity in deviations from the law of one price is equally provided.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 1 , February 2024 , pp. 60 - 97
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

We are grateful to Isaiah Andrews, Stéphane Bonhomme, Bo Honoré, Ryo Okui, and Peter Schmidt for comments, and to Arthur Lewbel and three referees for feedback on an earlier version. We also greatly appreciate the help of Ryo Okui and Mototsugu Shintani in providing us with the data used in our empirical illustration. Jochmans gratefully acknowledges financial support from the European Research Council through grant ERC-2016-StG-715787-MiMo and from the French Government and the ANR under the Investissements d’ Avenir program, grant ANR-17-EURE-0010. Weidner gratefully acknowledges support from the Economic and Social Research Council through the ESRC Centre for Microdata Methods and Practice grant RES-589-28-0001 and from the European Research Council grants ERC-2014-CoG-646917-ROMIA and ERC-2018-CoG-819086-PANEDA.

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