## The Annals of Statistics

### The linear stochastic order and directed inference for multivariate ordered distributions

#### Abstract

Researchers are often interested in drawing inferences regarding the order between two experimental groups on the basis of multivariate response data. Since standard multivariate methods are designed for two-sided alternatives, they may not be ideal for testing for order between two groups. In this article we introduce the notion of the linear stochastic order and investigate its properties. Statistical theory and methodology are developed to both estimate the direction which best separates two arbitrary ordered distributions and to test for order between the two groups. The new methodology generalizes Roy’s classical largest root test to the nonparametric setting and is applicable to random vectors with discrete and/or continuous components. The proposed methodology is illustrated using data obtained from a 90-day pre-chronic rodent cancer bioassay study conducted by the National Toxicology Program (NTP).

#### Article information

Source
Ann. Statist., Volume 41, Number 1 (2013), 1-40.

Dates
First available in Project Euclid: 5 March 2013

https://projecteuclid.org/euclid.aos/1362493038

Digital Object Identifier
doi:10.1214/12-AOS1062

Mathematical Reviews number (MathSciNet)
MR3059408

Zentralblatt MATH identifier
1266.60029

#### Citation

Davidov, Ori; Peddada, Shyamal. The linear stochastic order and directed inference for multivariate ordered distributions. Ann. Statist. 41 (2013), no. 1, 1--40. doi:10.1214/12-AOS1062. https://projecteuclid.org/euclid.aos/1362493038

#### References

• Abrevaya, J. and Huang, J. (2005). On the bootstrap of the maximum score estimator. Econometrica 73 1175–1204.
• Arcones, M. A., Kvam, P. H. and Samaniego, F. J. (2002). Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J. Amer. Statist. Assoc. 97 170–182.
• Audet, C., Béchard, V. and Le Digabel, S. (2008). Nonsmooth optimization through mesh adaptive direct search and variable neighborhood search. J. Global Optim. 41 299–318.
• Bekele, B. N. and Thall, P. F. (2004). Dose-finding based on multiple toxicities in a soft tissue sarcoma trial. J. Amer. Statist. Assoc. 99 26–35.
• Bickel, P. J. and Sakov, A. (2008). On the choice of $m$ in the $m$ out of $n$ bootstrap and confidence bounds for extrema. Statist. Sinica 18 967–985.
• DasGupta, A. (2008). Asymptotic Theory of Statistics and Probability. Springer, New York.
• Davey, B. A. and Priestley, H. A. (2002). Introduction to Lattices and Order, 2nd ed. Cambridge Univ. Press, New York.
• Davidov, O. (2012). Ordered inference, rank statistics and combining $p$-values: A new perspective. Stat. Methodol. 9 456–465.
• Davidov, O. and Herman, A. (2011). Multivariate stochastic orders induced by case-control sampling. Methodol. Comput. Appl. Probab. 13 139–154.
• Davidov, O. and Herman, A. (2012). Ordinal dominance curve based inference for stochastically ordered distributions. J. R. Stat. Soc. Ser. B Stat. Methodol. 74 825–847.
• Davidov, O. and Peddada, S. (2011). Order-restricted inference for multivariate binary data with application to toxicology. J. Amer. Statist. Assoc. 106 1394–1404.
• Delagdo, M., Rodriguez-Poo, J. M. and Wolf, M. (2001). Subsampling inference in cube root asymptotics with an application to Manski’s maximum score estimator. Econom. Lett. 73 241–250.
• Ding, Y. and Zhang, X. (2004). Some stochastic orders of Kotz-type distributions. Statist. Probab. Lett. 69 389–396.
• Fang, K. T., Kots, S. and Ng, K. W. (1989). Symmetric Multivariate and Related Distributions. Chapman & Hall, London.
• Fisher, N. I. and Hall, P. (1989). Bootstrap confidence regions for directional data. J. Amer. Statist. Assoc. 84 996–1002.
• Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests, 2nd ed. Academic Press, San Diego, CA.
• Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
• Hu, J., Homem-de-Mello, T. and Mehrotra, S. (2011). Concepts and applications of stochastically weighted stochastic dominance. Unpublished manuscript. Available at http://www.optimization-online.org/DB_FILE/2011/04/2981.pdf.
• Ivanova, A. and Murphy, M. (2009). An adaptive first in man dose-escalation study of NGX267: Statistical, clinical, and operational considerations. J. Biopharm. Statist. 19 247–255.
• Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.
• Johnson, R. and Wichern, D. (1998). Applied Multivariate Statistical Analysis. Prentice Hall, New York.
• Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219.
• Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer, New York.
• Lee, S. M. S. (1999). On a class of $m$ out of $n$ bootstrap confidence intervals. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 901–911.
• Lucas, L. A. and Wright, F. T. (1991). Testing for and against a stochastic ordering between multivariate multinomial populations. J. Multivariate Anal. 38 167–186.
• Moser, V. (2000). Observational batteries in neurotoxicity testing. International Journal of Toxicology 19 407–411.
• Neumeyer, N. (2004). A central limit theorem for two-sample $U$-processes. Statist. Probab. Lett. 67 73–85.
• NTP (2003). NTP toxicology and carcinogenesiss studies of citral (microencapsulated) (CAS No. 5392-40-5) in F344/N rats and B6C3F1 mice (feed studies).
• Peddada, S. D. (1985). A short note on Pitman’s measure of nearness. Amer. Statist. 39 298–299.
• Peddada, S. D. and Chang, T. (1996). Bootstrap confidence region estimation of the motion of rigid bodies. J. Amer. Statist. Assoc. 91 231–241.
• Pitman, E. J. G. (1937). The closest estimates of statistical parameters. Proceedings of the Cambridge Philosophical Society 33 212–222.
• Price, C. J., Reale, M. and Robertson, B. L. (2008). A direct search method for smooth and non-smooth unconstrained optimization problems. ANZIAM J. 48 927–948.
• Roy, S. N. (1953). On a heuristic method of test construction and its use in multivariate analysis. Ann. Math. Statist. 24 220–238.
• Sampson, A. R. and Whitaker, L. R. (1989). Estimation of multivariate distributions under stochastic ordering. J. Amer. Statist. Assoc. 84 541–548.
• Sen, B., Banerjee, M. and Woodroofe, M. (2010). Inconsistency of bootstrap: The Grenander estimator. Ann. Statist. 38 1953–1977.
• Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
• Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
• Sherman, R. P. (1993). The limiting distribution of the maximum rank correlation estimator. Econometrica 61 123–137.
• van der Vaart, A. W. (2000). Asymptotic Statistics. Cambridge Univ. Press, Cambridge.