Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T21:26:56.714Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected].
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

REFERENCES

Aït-Sahalia, Y. & Lo, A.W. (2000) Nonparametric risk management and implied risk aversion. Journal of Econometrics 94, 9–51.Google Scholar
Anderson, G. (1996) Nonparametric tests of stochastic dominance in income distributions. Econometrica 64, 1183–1193.Google Scholar
Barrett, G.F. & Donald, S.G. (2003) Consistent tests for stochastic dominance. Econometrica 71, 71–104.CrossRefGoogle Scholar
Beare, B.K. (2011) Measure preserving derivatives and the pricing kernel puzzle. Journal of Mathematical Economics 47, 689–697.CrossRefGoogle Scholar
Bera, A.K., Ghosh, A., & Xiao, Z. (2013) A smooth test for the equality of distributions. Econometric Theory 29, 419–446.Google Scholar
Bonnans, J.F. & Shapiro, A. (2000) Perturbation Analysis of Optimization Problems.Springer-Verlag.CrossRefGoogle Scholar
Carolan, C.A. (2002) The least concave majorant of the empirical distribution function. Canadian Journal of Statistics 30, 317–328.CrossRefGoogle Scholar
Carolan, C.A. & Tebbs, J.M. (2005) Nonparametric tests for and against likelihood ratio ordering in the two sample problem. Biometrika 92, 159–171.CrossRefGoogle Scholar
Davidson, R. & Duclos, J.-Y. (2000) Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica 68, 1435–1464.Google Scholar
Davidson, R. & Duclos, J.-Y. (2013) Testing for restricted stochastic dominance. Econometric Reviews 32, 84–125.CrossRefGoogle Scholar
Delgado, M.A. & Escanciano, J.C. (2012) Distribution-free tests of stochastic monotonicity. Journal of Econometrics 170, 68–75.CrossRefGoogle Scholar
Delgado, M.A. & Escanciano, J.C. (2013) Conditional stochastic dominance testing. Journal of Business and Economic Statistics 31, 16–28.Google Scholar
Durot, C. & Tocquet, A.-S. (2003) On the distance between the empirical process and its concave majorant in a monotone regression framework. Annales de l’Institut Henri Poincaré (B) 39, 217–240.CrossRefGoogle Scholar
Dykstra, R., Kochar, S., & Robertson, T. (1995) Inference for likelihood ratio ordering in the two-sample problem. Journal of the American Statistical Association 90, 1034–1040.Google Scholar
Hens, T. & Reichlin, C. (2013) Three solutions to the pricing kernel puzzle. Review of Finance 17, 1065–1098.CrossRefGoogle Scholar
Hsieh, F. & Turnbull, B.W. (1996) Nonparametric and semiparametric estimation of the receiver operating characteristic curve. Annals of Statistics 24, 25–40.Google Scholar
Jackwerth, J.C. (2000) Recovering risk aversion from option prices and realized returns. Review of Financial Studies 13, 433–451.CrossRefGoogle Scholar
Linton, O., Maasoumi, E., & Whang, Y.-J. (2005) Consistent testing for stochastic dominance under general sampling schemes. Review of Economic Studies 72, 735–765.Google Scholar
Linton, O., Song, K., & Whang, Y.-J. (2010) An improved bootstrap test of stochastic dominance. Journal of Econometrics 154, 186–202.Google Scholar
Maasoumi, E. (2001) Parametric and nonparametric tests of limited domain and ordered hypotheses in economics. In Baltagi, B. (ed.), A Companion to Econometric Theory. Blackwell.Google Scholar
Roosen, J. & Hennessy, D.A. (2004) Testing for the monotone likelihood ratio assumption. Journal of Business and Economic Statistics 28, 358–366.CrossRefGoogle Scholar
Rosenberg, J.V. & Engle, R.F. (2002) Empirical pricing kernels. Journal of Financial Economics 64, 341–372.CrossRefGoogle Scholar
Scaillet, O. & Topaloglou, N. (2010) Testing for stochastic dominance efficiency. Journal of Business and Economic Statistics 28, 169–180.CrossRefGoogle Scholar
Schechtman, E., Shelef, A., Shlomo, Y., & Zitikis, R. (2008) Testing hypotheses about absolute concentration curves and marginal conditional stochastic dominance. Econometric Theory 24, 1044–1062.Google Scholar
Shaked, M. & Shanthikumar, J. (1994) Stochastic Orders and their Applications.Academic Press.Google Scholar
Shapiro, A. (1990) On concepts of directional differentiability. Journal of Optimization Theory and Applications 66, 477–487.CrossRefGoogle Scholar
Shapiro, A. (1991) Asymptotic analysis of stochastic programs. Annals of Operations Research 30, 169–186.CrossRefGoogle Scholar
Thas, O. (2009) Comparing Distributions. Springer-Verlag.Google Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer-Verlag.CrossRefGoogle Scholar