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NONPARAMETRIC TESTS OF DENSITY RATIO ORDERING

Published online by Cambridge University Press:  08 September 2014

Brendan K. Beare  and
Jong-Myun Moon
Brendan K. Beare*
Affiliation:
University of California, San Diego
Jong-Myun Moon
Affiliation:
University of California, San Diego
*
*Address correspondence to Brendan Beare, Department of Economics, University of California - San Diego, 9500 Gilman Drive #0508, La Jolla, CA 92093-0508; email: bbeare@ucsd.edu.
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Abstract

We study a family of nonparametric tests of density ratio ordering between two continuous probability distributions on the real line. Density ratio ordering is satisfied when the two distributions admit a nonincreasing density ratio. Equivalently, density ratio ordering is satisfied when the ordinal dominance curve associated with the two distributions is concave. To test this property, we consider statistics based on the Lp-distance between an empirical ordinal dominance curve and its least concave majorant. We derive the limit distribution of these statistics when density ratio ordering is satisfied. Further, we establish that, when 1 ≤ p ≤ 2, the limit distribution is stochastically largest when the two distributions are equal. When 2 < p ≤ ∞, this is not the case, and in fact the limit distribution diverges to infinity along a suitably chosen sequence of concave ordinal dominance curves. Our results serve to clarify, extend, and amend assertions appearing previously in the literature for the cases p = 1 and p = ∞. We provide numerical evidence confirming their relevance in finite samples.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 3 , June 2015 , pp. 471 - 492
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

We thank Juan Carlos Escanciano, Hiroaki Kaido, Sokbae Lee, Andres Santos, Juwon Seo, Xiaoxia Shi, Joshua Tebbs, Yoon-Jae Whang, the anonymous referees, and seminar participants at Indiana University, Seoul National University, Sungkyunkwan University, the University of Cambridge, the University of Illinois at Urbana-Champaign, the University of Oxford, and Yonsei University for helpful comments.

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