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Nonparametric tests for stochastic ordering

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  • Teresa Ledwina

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  • Grzegorz Wyłupek
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    Abstract

    We present two new tests for stochastic ordering in a standard two-sample scheme. We approach the problem via its reparametrization in terms of Fourier coefficients in some corresponding system of functions and combining the resulting empirical Fourier coefficients. The empirical Fourier coefficients can be seen to be the asymptotically optimal linear rank statistics for the local sequences of nonparametric alternatives related to the introduced system of functions. Therefore, our first construction of the test is via multiple testing. The second test is based on sum of squares of censored empirical Fourier coefficients with the number of summands determined via a new model selection rule. The selection rule is fully automatic. Extensive simulations show that the new solutions improve upon existing tests based on adjusted variants of classical Kolmogorov–Smirnov, Anderson–Darling and L 1 -distance-based statistics, among others. We show that both tests control the error of the first kind for any fixed sample sizes and are capable of detecting essentially any alternative as the sample sizes are growing to infinity. We also discuss several aspects of our constructions, including possible efficiency calculations and asymptotic comparisons. Copyright Sociedad de Estadística e Investigación Operativa 2012

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    File URL: http://hdl.handle.net/10.1007/s11749-011-0278-7
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    Bibliographic Info

    Article provided by Springer in its journal TEST.

    Volume (Year): 21 (2012)
    Issue (Month): 4 (December)
    Pages: 730-756

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    Handle: RePEc:spr:testjl:v:21:y:2012:i:4:p:730-756

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    Related research

    Keywords: Data-driven test; Linear rank statistic; Model selection criteria; Multiple comparisons; Schwarz BIC; Stochastic dominance; Stochastic order; 62G10; 62G20;

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    1. Christensen, Ronald & Sun, Siu Kei, 2010. "Alternative Goodness-of-Fit Tests for Linear Models," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 291-301.
    2. Davidson, Russell & Duclos, Jean-Yves, 1998. "Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality," Cahiers de recherche 9805, Université Laval - Département d'économique.
    3. Juan Carlos Escanciano & Silvia Mayoral, . "Data-Driven Smooth Tests for the Martingale Difference Hypothesis," Faculty Working Papers 01/07, School of Economics and Business Administration, University of Navarra.
    4. Fan J. & Huang L-S., 2001. "Goodness-of-Fit Tests for Parametric Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 640-652, June.
    5. Anderson, Gordon, 1996. "Nonparametric Tests of Stochastic Dominance in Income Distributions," Econometrica, Econometric Society, vol. 64(5), pages 1183-93, September.
    6. Escanciano, J. Carlos & Lobato, Ignacio N., 2009. "An automatic Portmanteau test for serial correlation," Journal of Econometrics, Elsevier, vol. 151(2), pages 140-149, August.
    7. R. L. Eubank, 2000. "Testing for No Effect by Cosine Series Methods," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 27(4), pages 747-763.
    8. Garry F. Barrett & Stephen G. Donald, 2003. "Consistent Tests for Stochastic Dominance," Econometrica, Econometric Society, vol. 71(1), pages 71-104, January.
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    Cited by:
    1. Ledwina, Teresa & Wyłupek, Grzegorz, 2014. "Validation of positive quadrant dependence," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 38-47.

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