NONPARAMETRIC BAYESIAN MODELING FOR STOCHASTIC ORDER

ALAN E. GELFAND AND ATHANASIOS KOTTAS

Department of Statistics, The College of Liberal Arts and Sciences, University of Connecticut, 196 Auditorium Road, U-120, Storrs, CT 06269-3120, U.S.A.

(Received June 21, 1999; revised August 18, 2000)

Abstract. 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.

Key words and phrases: Dirichlet process mixing, linear functionals, Monte Carlo sampling and integration, semiparametric models.