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Statistical test for high order stochastic dominance under the density ratio model

  • 1.

    School of Management, University of Science and Technology of China, Hefei 230026, China

  • 2.

    School of Business, Xinhua University of Anhui, Hefei 230088, China

  • 3.

    International Institute of Finance, School of Management, University of Science and Technology of China, Hefei 230061, China

Cite this:
https://doi.org/10.52396/JUST-2021-0179
  • Received Date: 06 August 2021
  • Rev Recd Date: 30 September 2021
  • Publish Date: 31 December 2021
    Abstract

  • Abstract

    In economics, medicine and other fields, how to compare the dominance relations between two distributions has been widely discussed. Usually population means or medians are compared. However, the population with a higher mean may not be what we will choose, since it may also have a larger variance. Stochastic dominance proposes a good solution to this problem. Subsequently, how to test stochastic dominance relations between two distributions is worth discussing. In this paper, we develop the test statistic of high order stochastic dominance under the density ratio model. In addition, we provide the asymptotic properties of test statistic and use the bootstrap method to obtain p-values to make decisions. Furthermore, the simulation results show that the proposed test statistics have the high test power.

    Abstract

    In economics, medicine and other fields, how to compare the dominance relations between two distributions has been widely discussed. Usually population means or medians are compared. However, the population with a higher mean may not be what we will choose, since it may also have a larger variance. Stochastic dominance proposes a good solution to this problem. Subsequently, how to test stochastic dominance relations between two distributions is worth discussing. In this paper, we develop the test statistic of high order stochastic dominance under the density ratio model. In addition, we provide the asymptotic properties of test statistic and use the bootstrap method to obtain p-values to make decisions. Furthermore, the simulation results show that the proposed test statistics have the high test power.
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    • References(40)
  • [1]
    Lehmann E L. Testing Statistical Hypotheses. New York: Wiley, 1959.
    [2]
    Hadar J, Russell W R. Rules for ordering uncertain prospects. The American Economic Review, 1969, 59: 25-34.
    [3]
    Hanoch G, Levy H. The efficiency analysis of choices involving risk. The Review of Economic Studies, 1969, 36: 335-346.
    [4]
    Rothschild M, Stiglitz J E. Increasing risk: I.A definition. Journal of Economic Theory, 1970, 2: 225-243.
    [5]
    Anderson G. Nonparametric tests for stochastic dominance. Econometrica, 1996, 64: 1183-1193.
    [6]
    Davidson R, Duclos J Y. Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica, 2000, 68: 1435-1464.
    [7]
    McFadden D. Testing for Stochastic Dominance. New York: Springer, 1989.
    [8]
    Eubank R, Schechtman E, Yitzhaki S. A test for 2nd order stochastic dominance. Communications in Statistics: Theory and Methods, 1993, 22: 1893-1905.
    [9]
    Kaur A, Rao B L S, Singh H. Testing for 2nd-order stochastic dominance of 2 distributions. Econometric Theory, 1994, 10: 849-866.
    [10]
    Schmid F, Trede M. A Kolmogorov-type test for second-order stochastic dominance. Statistics and Probability Letters, 1998, 37: 183-193.
    [11]
    Barrett G F, Donald S G. Consistent tests for stochastic dominance. Econometrica, 2003, 71:71-104.
    [12]
    Donald S G, Hsu Y C. Improving the power of tests of stochastic dominance. Econometric Reviews, 2016, 35: 553-585.
    [13]
    Anderson J A. Multivariate logistic compounds. Biometrika, 1979, 66: 17-26.
    [14]
    Qin J, Zhang B. A goodness-of-t test for logistic regression models based on case-control data. Biometrika, 1997, 84: 609-618.
    [15]
    Keziou A, Leoni-Aubin S. On empirical likelihood for semiparametric two-sample density ratio models. Journal of Statistical Planning and Inference, 2008, 138: 915-928.
    [16]
    Chen J, Liu Y. Quantile and quantile-function estimations under density ratio model. The Annals of Statistics, 2013, 41: 1669-1692.
    [17]
    Qin J. Biased Sampling, Over-identified Parameter Problems and Beyond. New York: Springer, 2017.
    [18]
    Wang C, Marriott P, Li P. Testing homogeneity for multiple nonnegative distributions with excess zero observations. Computational Statistics and Data Analysis, 2017, 114: 146-157.
    [19]
    Wang C, Marriott P, Li P. Semiparametric inference on the means of multiple nonnegative distributions with excess zero observations. Journal of Multivariate Analysis, 2018, 166: 182-197.
    [20]
    Anderson J A. Separate sample logistic discrimination. Biometrika, 1972, 59: 19-35.
    [21]
    Anderson J A. Diagnosis by logistic discriminant function:Further practical problems and results. Journal of the Royal Statistical Society: Series C (Applied Statistics), 1974, 23: 397-404.
    [22]
    Owen A B. Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 1988, 75: 237-249.
    [23]
    Owen A B. Empirical likelihood ratio confidence regions. The Annals of Statistics, 1990, 18: 90-120.
    [24]
    Qin J. Inferences for case-control and semiparametric two-sample density ratio models. Biometrika, 1998, 85: 619-630.
    [25]
    Zhuang W, Hu B, Chen J. Semiparametric inference for the dominance index under the density ratio model. Biometrika, 2019, 106: 229-241.
    [26]
    Linton O, Song K, Whang Y J. An improved bootstrap test of stochastic dominance. Journal of Econometrics, 2010, 154: 186-202.
    [27]
    Owen A B. Empirical Likelihood. New York: Chapman and Hall, 2001.
    [28]
    Li H, Liu Y, Liu Y, et al. Comparison of empirical likelihood and its dual likelihood under density ratio model. Journal of Nonparametric Statistics, 2018, 30(3): 581-597.
    [29]
    Bishop J A, Chow K V, Formby J P. A stochastic dominance analysis of growth, recessions and the U.S. income distribution, 1967-1986. Southern Economic Journal, 1991, 57(4): 936-946.
    [30]
    Meyer J. Further applications of stochastic dominance to mutual fund performance. Journal of Financial and Quantitative Analysis, 1977, 12: 235-242.
    [31]
    Gasbarro D, Wong W K, Kenton-Zumwalt J. Stochastic dominance analysis of iShares. The European Journal of Finance, 2007, 13(1): 89-101.
    [32]
    Mann H B, Wald A. On the statistical treatment of linear stochastic difference equations. Econometrica, 1943, 11(3/4): 173-220.
    [33]
    Kay R, Little S. Transformations of the explanatory variables in the logistic regression model for binary data. Biometrika, 1987, 74: 495-501.
    [34]
    Kudo A. A multivariate analogue of one-sided tests. Biometrika, 1963, 50: 403-418.
    [35]
    Owen A. Empirical likelihood for linear models. The Annals of Statistics, 1991, 19: 1725-1747.
    [36]
    Pollard D. Convergence of Stochastic Processes. New York: Springer, 1984.
    [37]
    Randles R H, Wolfe D A. Introduction to the Theory of Nonparametric Statistics. New York: Wiley, 1979.
    [38]
    Shorack G R, Wellner J A. Empirical Processes with Applications in Statistics. New York: Wiley, 1986.
    [39]
    Van Der Vaart A W, Wellner J A. Weak Convergence and Empirical Processes with Applications to Statistics. New York: Springer, 1996.
    [40]
    Wolak F A. Testing inequality constraints in linear econometric models. Journal of Econometrics, 1989, 41: 205-235.
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    [1]
    Lehmann E L. Testing Statistical Hypotheses. New York: Wiley, 1959.
    [2]
    Hadar J, Russell W R. Rules for ordering uncertain prospects. The American Economic Review, 1969, 59: 25-34.
    [3]
    Hanoch G, Levy H. The efficiency analysis of choices involving risk. The Review of Economic Studies, 1969, 36: 335-346.
    [4]
    Rothschild M, Stiglitz J E. Increasing risk: I.A definition. Journal of Economic Theory, 1970, 2: 225-243.
    [5]
    Anderson G. Nonparametric tests for stochastic dominance. Econometrica, 1996, 64: 1183-1193.
    [6]
    Davidson R, Duclos J Y. Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica, 2000, 68: 1435-1464.
    [7]
    McFadden D. Testing for Stochastic Dominance. New York: Springer, 1989.
    [8]
    Eubank R, Schechtman E, Yitzhaki S. A test for 2nd order stochastic dominance. Communications in Statistics: Theory and Methods, 1993, 22: 1893-1905.
    [9]
    Kaur A, Rao B L S, Singh H. Testing for 2nd-order stochastic dominance of 2 distributions. Econometric Theory, 1994, 10: 849-866.
    [10]
    Schmid F, Trede M. A Kolmogorov-type test for second-order stochastic dominance. Statistics and Probability Letters, 1998, 37: 183-193.
    [11]
    Barrett G F, Donald S G. Consistent tests for stochastic dominance. Econometrica, 2003, 71:71-104.
    [12]
    Donald S G, Hsu Y C. Improving the power of tests of stochastic dominance. Econometric Reviews, 2016, 35: 553-585.
    [13]
    Anderson J A. Multivariate logistic compounds. Biometrika, 1979, 66: 17-26.
    [14]
    Qin J, Zhang B. A goodness-of-t test for logistic regression models based on case-control data. Biometrika, 1997, 84: 609-618.
    [15]
    Keziou A, Leoni-Aubin S. On empirical likelihood for semiparametric two-sample density ratio models. Journal of Statistical Planning and Inference, 2008, 138: 915-928.
    [16]
    Chen J, Liu Y. Quantile and quantile-function estimations under density ratio model. The Annals of Statistics, 2013, 41: 1669-1692.
    [17]
    Qin J. Biased Sampling, Over-identified Parameter Problems and Beyond. New York: Springer, 2017.
    [18]
    Wang C, Marriott P, Li P. Testing homogeneity for multiple nonnegative distributions with excess zero observations. Computational Statistics and Data Analysis, 2017, 114: 146-157.
    [19]
    Wang C, Marriott P, Li P. Semiparametric inference on the means of multiple nonnegative distributions with excess zero observations. Journal of Multivariate Analysis, 2018, 166: 182-197.
    [20]
    Anderson J A. Separate sample logistic discrimination. Biometrika, 1972, 59: 19-35.
    [21]
    Anderson J A. Diagnosis by logistic discriminant function:Further practical problems and results. Journal of the Royal Statistical Society: Series C (Applied Statistics), 1974, 23: 397-404.
    [22]
    Owen A B. Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 1988, 75: 237-249.
    [23]
    Owen A B. Empirical likelihood ratio confidence regions. The Annals of Statistics, 1990, 18: 90-120.
    [24]
    Qin J. Inferences for case-control and semiparametric two-sample density ratio models. Biometrika, 1998, 85: 619-630.
    [25]
    Zhuang W, Hu B, Chen J. Semiparametric inference for the dominance index under the density ratio model. Biometrika, 2019, 106: 229-241.
    [26]
    Linton O, Song K, Whang Y J. An improved bootstrap test of stochastic dominance. Journal of Econometrics, 2010, 154: 186-202.
    [27]
    Owen A B. Empirical Likelihood. New York: Chapman and Hall, 2001.
    [28]
    Li H, Liu Y, Liu Y, et al. Comparison of empirical likelihood and its dual likelihood under density ratio model. Journal of Nonparametric Statistics, 2018, 30(3): 581-597.
    [29]
    Bishop J A, Chow K V, Formby J P. A stochastic dominance analysis of growth, recessions and the U.S. income distribution, 1967-1986. Southern Economic Journal, 1991, 57(4): 936-946.
    [30]
    Meyer J. Further applications of stochastic dominance to mutual fund performance. Journal of Financial and Quantitative Analysis, 1977, 12: 235-242.
    [31]
    Gasbarro D, Wong W K, Kenton-Zumwalt J. Stochastic dominance analysis of iShares. The European Journal of Finance, 2007, 13(1): 89-101.
    [32]
    Mann H B, Wald A. On the statistical treatment of linear stochastic difference equations. Econometrica, 1943, 11(3/4): 173-220.
    [33]
    Kay R, Little S. Transformations of the explanatory variables in the logistic regression model for binary data. Biometrika, 1987, 74: 495-501.
    [34]
    Kudo A. A multivariate analogue of one-sided tests. Biometrika, 1963, 50: 403-418.
    [35]
    Owen A. Empirical likelihood for linear models. The Annals of Statistics, 1991, 19: 1725-1747.
    [36]
    Pollard D. Convergence of Stochastic Processes. New York: Springer, 1984.
    [37]
    Randles R H, Wolfe D A. Introduction to the Theory of Nonparametric Statistics. New York: Wiley, 1979.
    [38]
    Shorack G R, Wellner J A. Empirical Processes with Applications in Statistics. New York: Wiley, 1986.
    [39]
    Van Der Vaart A W, Wellner J A. Weak Convergence and Empirical Processes with Applications to Statistics. New York: Springer, 1996.
    [40]
    Wolak F A. Testing inequality constraints in linear econometric models. Journal of Econometrics, 1989, 41: 205-235.
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