SOME NONPARAMETRIC BAYESIAN ESTIMATION PROBLEMS.
Abstract
Two sample nonparametric decision problems for single parameter families of distributions are considered from the Bayesian viewpoint when only the relative magnitudes of the observation are known for the paired comparison data, the rank order data and the signed rank order data. The term nonparametric is used as the decision procedures depend on nonparametric statistics. It is assumed that the likelihood functions for the three kinds of data depend on the parameters of the sampled populations through some function H. A prior distribution is considered for the random variable H. The problem of the Bayes estimation of H, with the squared error loss function, is considered for the three kinds of data. A number of analytic properties of the posterior distribution of H and the orderings of the values of the Bayes estimate of H are obtained. Examples are given from a normal and a uniform family of distributions. Sufficient conditions are given on the sampled populations for the risk of the Bayes estimate of H to go to zero as the sample sizes go to infinity for the three kinds of data. The Bayes two decision problem, the two decisions being H > h and H < or = h, with the (0,1) loss function, is considered for the paired comparison data and the rank order data. Monotonic properties of the Bayes decision procedures are obtained under mild conditions on the sampled populations. Sufficient conditions on the sampled populations are given for the risk of the Bayes two decision procedures to go to zero as the sample sizes go to infinity. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1965
- Accession Number
- AD0623834
Entities
People
- K. M. Lal Saxena
Organizations
- Florida State University