Nonparametric Bayesian Modeling for Stochastic Order

  • Alan E. Gelfand
  • Athanasios Kottas


In comparing two populations, sometimes a model incorporating stochastic order is desired. Customarily, such modeling is done parametrically. The objective of this paper is to formulate nonparametric (possibly semiparametric) stochastic order specifications providing richer, more flexible modeling. We adopt a fully Bayesian approach using Dirichlet process mixing. An attractive feature of the Bayesian approach is that full inference is available regarding the population distributions. Prior information can conveniently be incorporated. Also, prior stochastic order is preserved to the posterior analysis. Apart from the two sample setting, the approach handles the matched pairs problem, the k-sample slippage problem, ordered ANOVA and ordered regression models. We illustrate by comparing two rather small samples, one of diabetic men, the other of diabetic women. Measurements are of androstenedione levels. Males are anticipated to produce levels which will tend to be higher than those of females.

Dirichlet process mixing linear functionals Monte Carlo sampling and integration semiparametric models 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akritas, M. G. and Arnold, S. F. (1994). Fully nonparametric hypotheses for factorial designs I: Multivariate repeated measures designs, J. Amer. Statist. Assoc., 89, 336-343.Google Scholar
  2. Antoniak, C. E. (1974). Mixtures of dirichlet processes with applications to nonparametric problems, Ann. Statist., 2, 1152-1174.Google Scholar
  3. Arjas, E., and Gasbarra, D. (1996). Bayesian inference of survival probabilities, under stochastic ordering constraints, J. Amer. Statist. Assoc., 91, 1101-1109.Google Scholar
  4. Devroye, L. (1986). Non-uniform Random Variate Generation, Springer, New York.Google Scholar
  5. Escobar, M. D., and West, M. (1995). Bayesian density estimation and inference using mixtures, J. Amer. Statist. Assoc., 90, 577-588.Google Scholar
  6. Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems, Ann. Statist., 1, 209-230.Google Scholar
  7. Gelfand, A. E., and Kottas, A. (2001). A computational approach for full nonparametric Bayesian inference in single and multiple sample problems, J. Comput. Graph. Statist. (to appear).Google Scholar
  8. Gelfand, A. E., and Mukhopadhyay, S. (1995). On nonparametric Bayesian inference for the distribution of a random sample, Canad. J. Statist., 23, 411-420.Google Scholar
  9. Gelfand, A. E., and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities, J. Amer. Statist. Assoc., 85, 398-409.Google Scholar
  10. Joe, H. (1997). Multivariate Models and Dependence Concepts, Chapman and Hall, London.Google Scholar
  11. Koopmans, L. H. (1987). Introduction to Contemporary Statistical Methods, Duxbury, Belmont, California.Google Scholar
  12. Lehmann, E. (1986). Testing Statistical Hypotheses, 2nd ed., Wiley, New York.Google Scholar
  13. Lo, A. Y. (1984). On a class of bayesian nonparametric estimates: I. Density estimates, Ann. Statist., 12, 351-357.Google Scholar
  14. MacEachern, S. N., and Müller, P. (1998). Estimating mixture of Dirichlet process Models, J. Comput. Graph. Statist., 7, 223-238.Google Scholar
  15. Mukhopadhyay, S., and Gelfand, A. E. (1997). Dirichlet process mixed generalized linear models, J. Amer. Statist. Assoc., 92, 633-639.Google Scholar
  16. Randles, R. H., and Wolfe, D. A. (1979). Introduction to The Theory of Nonparametric Statistics, Wiley, New York.Google Scholar
  17. Sethuraman, J. (1994). A constructive definition of Dirichlet priors, Statistica Sinica, 4, 639-650.Google Scholar
  18. Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications, Academic Press, Boston.Google Scholar
  19. Walker, S. G., and Damien, P. (1998). Sampling Methods for Bayesian Nonparametric Inference Involving Stochastic Processes, Practical Nonparametric and Semiparametric Bayesian Statistics (eds. D. Dey, P. Müller and D. Sinha), 243-254, Springer, New York.Google Scholar
  20. Walker, S. G., Damien, P., Laud, P. W., and Smith, A. F. M. (1999). Bayesian nonparametric inference for random distributions and related functions (with discussion), J. Roy. Statist. Soc. Ser. B, 61, 485-527.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 2001

Authors and Affiliations

  • Alan E. Gelfand
    • 1
  • Athanasios Kottas
    • 1
  1. 1.Department of Statistics, The College of Liberal Arts and SciencesUniversity of ConnecticutStorrsUSA

Personalised recommendations