An improved bootstrap test of density ratio ordering

beare, brendan and shi, xiaoxia (2015): An improved bootstrap test of density ratio ordering.

Abstract

Two probability distributions with common support are said to exhibit density ratio ordering when they admit a nonincreasing density ratio. Existing statistical tests of the null hypothesis of density ratio ordering are known to be conservative, with null limiting rejection rates below the nominal significance level whenever the two distributions are unequal. We show how a bootstrap procedure can be used to shrink the critical values used in existing procedures such that the limiting rejection rate is increased to the nominal significance level on the boundary of the null. This improves power against nearby alternatives. Our procedure is based on preliminary estimation of a contact set, the form of which is obtained from a novel representation of the Hadamard directional derivative of the least concave majorant operator. Numerical simulations indicate that improvements to power can be very large in moderately sized samples.

• Item Type: MPRA Paper
• Language: English
• Keywords: bootstrap, density ratio ordering, power
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General

References

Anderson, G. (1996). Nonparametric tests of stochastic dominance in income distributions. Econometrica 64 1183–1193.

Anderson, G., Linton, O. and Whang, Y. -J. (2012). Nonparametric estimation and inference about the overlap of two distributions. Journal of Econometrics 171 1–23.

Andrews, D. W. K. and Shi, X. (2013). Inference based on conditional moment inequalities. Econometrica 81 609–666.

Barrett, G. F. and Donald, S. G. (2003). Consistent tests for stochastic dominance. Econometrica 71 71–104.

Beare, B. K. (2011). Measure preserving derivatives and the pricing kernel puzzle. Journal of Mathematical Economics 47 689–697.

Beare, B. K. and Moon, J. -M. (2015). Nonparametric tests of density ratio ordering. Econometric Theory 31 471–492.

Beare, B. K. and Schmidt, L. D. W. (2015). An empirical test of pricing kernel monotonicity. Journal of Applied Econometrics, in press.

Beutner, E., Wu, W. B. and Zahle, H. ¨ (2012). Asymptotics for statistical functionals of long-memory sequences. Stochastic Processes and their applications 122, 910–929.

Beutner, E. and Zahle, H. ¨ (2010). A modified functional delta method and its application to the estimation of risk functionals. Journal of Multivariate Analysis 101, 2452–2463.

Beutner, E. and Zahle, H. ¨ (2012). Deriving the asymptotic distribution of u- and vstatistics of dependent data using weighted empirical processes. Bernoulli 18, 803–822.

Carolan, C. A. (2002). The least concave majorant of the empirical distribution function. Canadian Journal of Statistics 30 317–328.

Carolan, C. A. and Tebbs, J. M. (2005). Nonparametric tests for and against likelihood ratio ordering in the two sample problem. Biometrika 92 159–171.

Davidson, R. and Duclos, J. -Y. (2000). Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica 68 1435–1464.

Delgado, M. A. and Escanciano, J. C. (2012). Distribution-free tests of stochastic monotonicity. Journal of Econometrics 170 68–75.

Delgado, M. A. and Escanciano, J. C. (2013). Conditional stochastic dominance testing. Journal of Business and Economic Statistics 31 16–28.

Donald, S. G. and Hsu, Y. -C. (2015). Improving the power of tests of stochastic dominance. Econometric Reviews, in press.

Dumbgen, L. ¨ (1993). On nondifferentiable functions and the bootstrap. Probability Theory and Related Fields 95 125–140.

Dykstra, R., Kochar, S. and Robertson, T. (1995). Inference for likelihood ratio ordering in the two-sample problem. Journal of the American Statistical Association 90 1034–1040.

Fang, Z. and Santos, A. (2014). Inference on directionally differentiable functions. Preprint, UC San Diego.

Giacomini, R., Politis, D. N. and White, H. (2013). A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econometric Theory 29 567–589.

Hsieh, F. and Turnbull, B. W. (1996). Nonparametric and semiparametric estimation of the receiver operating characteristic curve. Annals of Statistics 24 25–40.

Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics, New York.

Lee, S. and Whang, Y. -J. (2009). Nonparametric tests of conditional treatment effects. CeMMAP Working Paper No. 36/09, Institute for Fiscal Studies.

Lee, S., Song, K. and Whang, Y. -J. (2014). Testing for a general class of functional inequalities. CeMMAP Working Paper No. 09/14, Institute for Fiscal Studies.

Linton, O., Maasoumi, E. and Whang, Y. -J. (2005). Consistent testing for stochastic dominance under general sampling schemes. Review of Economic Studies 72 735–765.

Linton, O., Song, K. and Whang, Y. -J. (2010). An improved bootstrap test of stochastic dominance. Journal of Econometrics 154 186–202.

Maasoumi, E. (2001). Parametric and nonparametric tests of limited domain and ordered hypotheses in economics. In Baltagi, B. (Ed.), A Companion to Econometric Theory. Blackwell Pub., Oxford.

Roosen, J. and Hennessy, D. A. (2004). Testing for the monotone likelihood ratio assumption. Journal of Business and Economic Statistics 28 358–366. Seo, J. (2014). Tests of stochastic monotonicity with improved size and power properties. 28

Shapiro, A. (1990). On concepts of directional differentiability. Journal of Optimization Theory and Applications 66 477–487.

Shapiro, A. (1991). Asymptotic analysis of stochastic programs. Annals of Operations Research 30 169–186.

Volgushev, S. and Shao, X. (2014). A general approach to the joint asymptotic analysis of statistics from subsamples. Electronic Journal of Statistics 8, 390–431.