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A contamination model for the stochastic order

Author

Listed:
  • P. C. Álvarez-Esteban

    (Universidad de Valladolid)

  • E. del Barrio

    (Universidad de Valladolid)

  • J. A. Cuesta-Albertos

    (Universidad de Cantabria)

  • C. Matrán

    (Universidad de Valladolid)

Abstract

Stochastic ordering among distributions has been considered in a variety of scenarios. However, it is often a restrictive model, not supported by the data even in cases in which the researcher tends to believe that a certain variable is somehow smaller than other. Alternatively, we propose to look at a more flexible version in which two distributions satisfy an approximate stochastic order relation if they are slightly contaminated versions of distributions for which stochastic order holds. The minimal level of contamination required for stochastic order to hold is used as a measure of deviation from exact stochastic order model. Our approach is based on the use of trimmings of probabilities. We discuss their connection to approximate stochastic order and provide theoretical support for its use in data analysis, proving uniform consistency and giving non-asymptotic bounds for the error probabilities of our tests. We provide simulation results and a case study for illustration.

Suggested Citation

  • P. C. Álvarez-Esteban & E. del Barrio & J. A. Cuesta-Albertos & C. Matrán, 2016. "A contamination model for the stochastic order," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 751-774, December.
    • Handle: RePEc:spr:testjl:v:25:y:2016:i:4:d:10.1007_s11749-016-0494-2
    DOI: 10.1007/s11749-016-0494-2
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    References listed on IDEAS

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    1. Russell Davidson & Jean-Yves Duclos, 2013. "Testing for Restricted Stochastic Dominance," Econometric Reviews, Taylor & Francis Journals, vol. 32(1), pages 84-125, January.
    2. 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.
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    4. Russell Davidson & Jean-Yves Duclos, 2000. "Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality," Econometrica, Econometric Society, vol. 68(6), pages 1435-1464, November.
    5. Garry F. Barrett & Stephen G. Donald, 2003. "Consistent Tests for Stochastic Dominance," Econometrica, Econometric Society, vol. 71(1), pages 71-104, January.
    6. Alvarez-Esteban, Pedro Cesar & del Barrio, Eustasio & Cuesta-Albertos, Juan Antonio & Matran, Carlos, 2008. "Trimmed Comparison of Distributions," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 697-704, June.
    7. Anderson, Gordon, 1996. "Nonparametric Tests of Stochastic Dominance in Income Distributions," Econometrica, Econometric Society, vol. 64(5), pages 1183-1193, September.
    8. Arcones M. A. & Kvam P. H. & Samaniego F. J., 2002. "Nonparametric Estimation of a Distribution Subject to a Stochastic Precedence Constraint," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 170-182, March.
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    Cited by:

    1. Gordon Anderson & Maria Grazia Pittau & Roberto Zelli & Jasmin Thomas, 2018. "Income Inequality, Cohesiveness and Commonality in the Euro Area: A Semi-Parametric Boundary-Free Analysis," Econometrics, MDPI, vol. 6(2), pages 1-20, March.
    2. Eustasio del Barrio & Hristo Inouzhe & Carlos Matrán, 2020. "Box-Constrained Monotone Approximations to Lipschitz Regularizations, with Applications to Robust Testing," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 65-87, October.
    3. Montes, Ignacio & Salamanca, Juan Jesús & Montes, Susana, 2020. "A modified version of stochastic dominance involving dependence," Statistics & Probability Letters, Elsevier, vol. 165(C).
    4. E. Barrio & H. Inouzhe & C. Matrán, 2020. "On approximate validation of models: a Kolmogorov–Smirnov-based approach," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 938-965, December.

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