A contamination model for the stochastic order. (English) Zbl 1373.60040
Summary: 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.
MSC:
| 60E15 | Inequalities; stochastic orderings |
| 62G10 | Nonparametric hypothesis testing |
| 62G35 | Nonparametric robustness |
Keywords:
stochastic order; contamination model; trimmed distribution; uniformly consistent test; asymptotics; bootstrapReferences:
| [1] | Álvarez-Esteban PC, del Barrio E, Cuesta-Albertos JA, Matrán C (2008) Trimmed comparison of distributions. J Am Stat Assoc 103:697-704 · Zbl 1471.62262 · doi:10.1198/016214508000000274 |
| [2] | Álvarez-Esteban PC, del Barrio E, Cuesta-Albertos JA, Matrán C (2011) Uniqueness and approximate computation of optimal incomplete transportation plans. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 47:358-375 · Zbl 1215.49042 · doi:10.1214/09-AIHP354 |
| [3] | Álvarez-Esteban PC, del Barrio E, Cuesta-Albertos JA, Matrán C (2012) Similarity of samples and trimming. Bernoulli 18:606-634 · Zbl 1239.62005 · doi:10.3150/11-BEJ351 |
| [4] | Álvarez-Esteban PC, del Barrio E, Cuesta-Albertos JA, Matrán C (2014) A contamination model for approximate stochastic order: extended version. http://arxiv.org/abs/1412.1920 · Zbl 1373.60040 |
| [5] | Anderson G (1996) Nonparametric tests for stochastic dominance. Econometrica 64:1183-1193 · Zbl 0856.90024 · doi:10.2307/2171961 |
| [6] | Arcones MA, Kvam PH, Samaniego FJ (2002) Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J Am Stat Assoc 97:170-182 · Zbl 1073.62520 · doi:10.1198/016214502753479310 |
| [7] | Barrett GF, Donald SG (2003) Consistent tests for stochastic dominance. Econometrica 71:71-104 · Zbl 1137.62332 · doi:10.1111/1468-0262.00390 |
| [8] | Barron AR (1989) Uniformly powerful goodness of fit tests. Ann Stat 17:107-124 · Zbl 0674.62032 · doi:10.1214/aos/1176347005 |
| [9] | Berger, RL; Gupta, SS (ed.); Berger, JO (ed.), A nonparametric, intersection-union test for stochastic order, No. 2 (1988), New York · Zbl 0687.62038 |
| [10] | Davidson R, Duclos JY (2000) Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica 68:1435-1464 · Zbl 1055.91543 · doi:10.1111/1468-0262.00167 |
| [11] | Davidson R, Duclos JY (2013) Testing for restricted stochastic dominance. Econ Rev 32:84-125 · Zbl 1491.62199 · doi:10.1080/07474938.2012.690332 |
| [12] | Gordaliza A (1991) Best approximations to random variables based on trimming procedures. J Approx Theor 64:162-180 · Zbl 0745.41030 · doi:10.1016/0021-9045(91)90072-I |
| [13] | Hodges JL, Lehmann EL (1954) Testing the approximate validity of statistical hypotheses. J R Stat Soc B 16:261-268 · Zbl 0057.35403 |
| [14] | Lehmann EL (1955) Ordered families of distributions. Ann Math Stat 26:399-419 · Zbl 0065.11906 · doi:10.1214/aoms/1177728487 |
| [15] | Lindsay BG, Liu J (2009) Model assessment tools for a model false world. Stat Sci 24:303-318 · Zbl 1329.62099 · doi:10.1214/09-STS302 |
| [16] | Linton O, Maasoumi E, Whang YJ (2005) Consistent testing for stochastic dominance under general sampling schemes. Rev Econ Stud 72:735-765 · Zbl 1070.62031 · doi:10.1111/j.1467-937X.2005.00350.x |
| [17] | Linton O, Song K, Whang YJ (2010) An improved bootstrap test for stochastic dominance. J Econ 154:186-202 · Zbl 1431.62190 · doi:10.1016/j.jeconom.2009.08.002 |
| [18] | Liu J, Lindsay BG (2009) Building and using semiparametric tolerance regions for parametric multinomial models. Ann Stat 37:3644-3659 · Zbl 1369.62100 · doi:10.1214/08-AOS603 |
| [19] | McFadden, D.; Fomby, TB (ed.); Seo, TK (ed.), Testing for stochastic dominance (1989), In Honor of Josef Hadar |
| [20] | Mosler, K.; Rinne, H. (ed.); Rüger, B. (ed.); Strecker, H. (ed.), Testing whether two distributions are stochastically ordered or not (1995), Heidelberg · Zbl 0851.62030 |
| [21] | Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, Chichester · Zbl 0999.60002 |
| [22] | Raghavachari M (1973) Limiting distributions of the Kolmogorov-Smirnov type statistics under the alternative. Ann Stat 1:67-73 · Zbl 0276.62028 · doi:10.1214/aos/1193342382 |
| [23] | Rudas T, Clogg CC, Lindsay BG (1994) A New Index of Fit Based on Mixture Methods for the Analysis of Contingency Tables. J R Stat Soc B 56:623-639 · Zbl 0800.62301 |
| [24] | Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York · Zbl 0806.62009 · doi:10.1007/978-0-387-34675-5 |
| [25] | Schmid F, Trede M (1996) Testing for first order stochastic dominance in either direction. Comput Stat 11:165-173 · Zbl 0935.62056 |
| [26] | Schmid F, Trede M (1996) Testing for first-order stochastic dominance: a new distribution-free test. Statistician 45:371-380 · Zbl 0935.62056 · doi:10.2307/2988473 |
| [27] | Shorack GR, Wellner JA (1986) Empirical processes with applications to statistics. Wiley · Zbl 1170.62365 |
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