IDEAS home Printed from https://ideas.repec.org/p/umc/wpaper/1801.html
   My bibliography  Save this paper

Comparing distributions by multiple testing across quantiles or CDF values

Author

Abstract

When comparing two distributions, it is often helpful to learn at which quantiles or values there is a statistically significant difference. This provides more information than the binary "reject" or "do not reject" decision of a global goodness-of-fit test. Framing our question as multiple testing across the continuum of quantiles tau in (0,1) or values r, we show that the Kolmogorov–Smirnov test (interpreted as a multiple testing procedure) achieves strong control of the familywise error rate. However, its well-known flaw of low sensitivity in the tails remains. We provide an alternative method that retains such strong control of familywise error rate while also having even sensitivity, i.e., equal pointwise type I error rates at each of n (going to infinity) order statistics across the distribution. Our one-sample method computes instantly, using our new formula that also instantly computes goodness-of-fit p-values and uniform confidence bands. To improve power, we also propose stepdown and pre-test procedures that maintain control of the asymptotic familywise error rate. One-sample and two-sample cases are considered, as well as extensions to regression discontinuity designs and conditional distributions. Simulations, empirical examples, and code are provided. Classification-JEL: C12, C14, C21

Suggested Citation

  • David M. Kaplan & Matt Goldman, 2013. "Comparing distributions by multiple testing across quantiles or CDF values," Working Papers 18-01, Department of Economics, University of Missouri, revised 22 Feb 2018.
  • Handle: RePEc:umc:wpaper:1801
    Note: Title change on 2018-02-22
    as

    Download full text from publisher

    File URL: https://economics.missouri.edu/sites/default/files/wp-files/gk2018_dist_inf.pdf
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    as
    1. Linton, Oliver & Song, Kyungchul & Whang, Yoon-Jae, 2010. "An improved bootstrap test of stochastic dominance," Journal of Econometrics, Elsevier, vol. 154(2), pages 186-202, February.
    2. Marianne P. Bitler & Jonah B. Gelbach & Hilary W. Hoynes, 2006. "What Mean Impacts Miss: Distributional Effects of Welfare Reform Experiments," American Economic Review, American Economic Association, vol. 96(4), pages 988-1012, September.
    3. A. Colin Cameron & Jonah B. Gelbach & Douglas L. Miller, 2008. "Bootstrap-Based Improvements for Inference with Clustered Errors," The Review of Economics and Statistics, MIT Press, vol. 90(3), pages 414-427, August.
    4. Russell Davidson & Jean-Yves Duclos, 2013. "Testing for Restricted Stochastic Dominance," Econometric Reviews, Taylor & Francis Journals, vol. 32(1), pages 84-125, January.
    5. Jackson, Erika & Page, Marianne E., 2013. "Estimating the distributional effects of education reforms: A look at Project STAR," Economics of Education Review, Elsevier, vol. 32(C), pages 92-103.
    6. Goldman, Matt & Kaplan, David M., 2017. "Fractional order statistic approximation for nonparametric conditional quantile inference," Journal of Econometrics, Elsevier, vol. 196(2), pages 331-346.
    7. R'emy Chicheportiche & Jean-Philippe Bouchaud, 2012. "Weighted Kolmogorov-Smirnov test: Accounting for the tails," Papers 1207.7308, arXiv.org, revised Oct 2012.
    8. David M. Kaplan & Matt Goldman, 2011. "Nonparametric inference on conditional quantile differences and linear combinations, using L-statistics," Working Papers 1503, Department of Economics, University of Missouri, revised 21 Nov 2016.
    9. Thomas MaCurdy & Xiaohong Chen & Han Hong, 2011. "Flexible Estimation of Treatment Effect Parameters," American Economic Review, American Economic Association, vol. 101(3), pages 544-551, May.
    10. David M. Kaplan & Longhao Zhuo, 2015. "Frequentist size of Bayesian inequality tests," Working Papers 1709, Department of Economics, University of Missouri, revised 26 Feb 2018.
    11. David M. Kaplan & Longhao Zhuo, 2015. "Frequentist size of Bayesian inequality tests," Working Papers 1709, Department of Economics, University of Missouri, revised 26 Feb 2018.
    12. Shu Shen & Xiaohan Zhang, 2016. "Distributional Tests for Regression Discontinuity: Theory and Empirical Examples," The Review of Economics and Statistics, MIT Press, vol. 98(4), pages 685-700, October.
    13. Bitler, Marianne P. & Gelbach, Jonah B. & Hoynes, Hilary W., 2008. "Distributional impacts of the Self-Sufficiency Project," Journal of Public Economics, Elsevier, vol. 92(3-4), pages 748-765, April.
    14. Djebbari, Habiba & Smith, Jeffrey, 2008. "Heterogeneous impacts in PROGRESA," Journal of Econometrics, Elsevier, vol. 145(1-2), pages 64-80, July.
    15. Susan Athey & Guido W. Imbens, 2006. "Identification and Inference in Nonlinear Difference-in-Differences Models," Econometrica, Econometric Society, vol. 74(2), pages 431-497, March.
    16. Moscovich, Amit & Nadler, Boaz, 2017. "Fast calculation of boundary crossing probabilities for Poisson processes," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 177-182.
    17. Qu, Zhongjun & Yoon, Jungmo, 2015. "Nonparametric estimation and inference on conditional quantile processes," Journal of Econometrics, Elsevier, vol. 185(1), pages 1-19.
    18. Ivan A. Canay & Vishal Kamat, 2015. "Approximate permutation tests and induced order statistics in the regression discontinuity design," CeMMAP working papers CWP27/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    19. Uri Gneezy & John A List, 2006. "Putting Behavioral Economics to Work: Testing for Gift Exchange in Labor Markets Using Field Experiments," Econometrica, Econometric Society, vol. 74(5), pages 1365-1384, September.
    20. Stephen G. Donald & Yu-Chin Hsu, 2016. "Improving the Power of Tests of Stochastic Dominance," Econometric Reviews, Taylor & Francis Journals, vol. 35(4), pages 553-585, April.
    21. Sivan Aldor-Noiman & Lawrence D. Brown & Andreas Buja & Wolfgang Rolke & Robert A. Stine, 2013. "The Power to See: A New Graphical Test of Normality," The American Statistician, Taylor & Francis Journals, vol. 67(4), pages 249-260, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. David M. Kaplan & Longhao Zhuo, 2015. "Frequentist size of Bayesian inequality tests," Working Papers 1802, Department of Economics, University of Missouri, revised 26 Feb 2018.

    More about this item

    Keywords

    Dirichlet; familywise error rate; Kolmogorov–Smirnov; probability integral transform; stepdown;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:umc:wpaper:1801. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Valerie Kulp) or (Ilker Cakar). General contact details of provider: http://edirc.repec.org/data/edumous.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.