Bayesian Estimation of Uniformly Stochastically Ordered Distributions with Square Loss

- Journal title : Communications for Statistical Applications and Methods
- Volume 18, Issue 3, 2011, pp.295-300
- Publisher : The Korean Statistical Society
- DOI : 10.5351/CKSS.2011.18.3.295

Title & Authors

Bayesian Estimation of Uniformly Stochastically Ordered Distributions with Square Loss

Oh, Myong-Sik;

Oh, Myong-Sik;

Abstract

The Bayesian nonparametric estimation of two uniformly stochastically ordered distributions is studied. We propose a restricted Dirichlet Process. Among many types of restriction we consider only uniformly stochastic ordering in this paper since the computation of integrals is relatively easy. An explicit expression of the posterior distribution is given. When square loss function is used the posterior distribution can be obtained by easy integration using some computer program such as Mathematica.

Keywords

Bayesian estimation;Dirichlet Process;square loss;uniform stochastic ordering;

Language

English

References

1.

Antoniak, C. E. (1974). Mixtures of Dirichlet processes with application to Bayesian nonparametric problems, Annals of Statistics, 2, 1152-1174.

2.

Doksum, K. (1974). Tail free and neutral random probabilities and their posterior distributions, Annals of Probability, 2, 183-201.

3.

Dunson, D. B. and Peddada, S. D. (2008). Bayesian nonparametric inference on stochastic ordering, Biometrika, 95, 859-874.

4.

Dykstra, R. L., Kochar, S. C. and Robertson, T. (1991). Statistical inference for uniform stochastic ordering in several populations, Annals of Statistics, 19, 870-880.

5.

Dykstra, R. L. and Laud, P. (1981). A Bayesian nonparametric approach to reliability, Annals of Statistics, 9, 356-367.

6.

Evans, M., Gilula, Z., Guttman, I. and Swartz, T. (1997). Bayesian analysis of stochastically ordered distributions of categorical variables, Journal of the American Statistical Association, 92, 208-214.

7.

Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems, Annals of Statistics, 1, 209-230.

8.

Ferguson, T. S. and Phadia, E. G. (1979). Bayesian nonparametric estimation based on censored data, Annals of Statistics, 7, 163-186.

9.

Hoff, P. D. (2003). Bayesian methods for partial stochastic orderings, Biometrika, 90, 203-317.

10.

Karabatsos, G. and Walker, S. G. (2009). Bayesian nonparametric inference of stochastically ordered distributions, with P´olya trees and Bernstein polynomials, Statistics and Probability Letters, 77, 907-913.

11.

Mukerjee, H. (1996). Estimation of survival functions under uniform stochastic ordering, Journal of the American Statistical Association, 91, 1684-1689.

12.

Proschan, F. and Singpurwalla, N. (1980). A new approach to inference from accerlated life tests, IEEE Transaction on Reliability, 29, 98-102.

13.

Rojo, J. and Samaniego, F. J. (1991). On nonparametric maximum likelihood estimation of a distribution uniformly stochastically smaller than a standard, Statistics and Probability Letters, 11, 267-271.