A class of neutral to the right priors induced by superposition of beta processes

A random distribution function on the positive real line which belongs to the class of neutral to the right priors is defined. It corresponds to the superposition of independent beta processes at the cumulative hazard level. The definition is constructive and starts with a discrete time process with random probability masses obtained from suitably defined products of independent beta random variables. The continuous time version is derived as the corresponding infinitesimal weak limit and is described in terms of completely random measures. It takes the interpretation of the survival distribution resulting from independent competing failure times. We discuss prior specification and illustrate posterior inference on a real data example.     

A Note on the Prior Distributions of Weibull Parameters for the Reliability Function

In Bayesian Inference it is often desirable to have a posterior density reflecting mainly the information from sample data. To achieve this purpose it is important to employ prior densities which add little information to the sample. We have in the literature many such prior densities, for example, Jeffreys (1967 Jeffreys , H. ( 1967 ). Theory of Probability , 3rd rev. ed. . London : Oxford University Press . [Google Scholar]), Lindley (1956 Lindley , D. V. ( 1956 ). On a measure of the information provided by an experiment . Ann. Mathemat. Statist. 27 : 986 – 1005 .[Crossref] , [Google Scholar]); (1961 Lindley , D. V. ( 1961 ). The use of prior probability distributions in statistical inference and decisions . In: Neyman , J. , ed. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability . Vol. 1. Berkeley : University of California Press , pp. 453 – 468 . [Google Scholar]), Hartigan (1964 Hartigan , J. ( 1964 ). Invariant priors distributions . Ann. Mathemat. Statist. 35 : 836 – 845 .[Crossref] , [Google Scholar]), Bernardo (1979 Bernardo , J. M. ( 1979 ). Reference posterior distributions for Bayesian inference . J. Roy. Statist. Soc. 41 ( 2 ): 113 – 147 . [Google Scholar]), Zellner (1984 Zellner , A. ( 1984 ). Maximal Data Information Prior Distributions, Basic Issues in Econometrics . Chicago : University of Chicago Press . [Google Scholar]), Tibshirani (1989 Tibshirani , R. ( 1989 ). Noninformative priors for one parameters of many . Biometrika 76 : 604 – 608 .[Crossref], [Web of Science ®] , [Google Scholar]), etc. In the present article, we compare the posterior densities of the reliability function by using Jeffreys, the maximal data information (Zellner, 1984 Zellner , A. ( 1984 ). Maximal Data Information Prior Distributions, Basic Issues in Econometrics . Chicago : University of Chicago Press . [Google Scholar]), Tibshirani's, and reference priors for the reliability function R(t) in a Weibull distribution.     

Analysis of Frechet Distribution Using Reference Priors

This article develops the Bayesian estimators in the context of reference priors for the two-parameter Frechet distribution. The general forms of the second-order matching priors are also derived in case of any parameter of interest and concluded that the reference prior is also a second order matching prior. Since the Bayesian estimators cannot be obtained in closed form, they are obtained using Monte Carlo simulation and Laplace approximation. The Bayesian and maximum likelihood estimates are compared via simulation study. Two real-life data sets are analyzed for illustration and comparison purpose.     

An examination of distributed lag model coefficients estimated with smoothness priors

The estimator for the coefficients of a distributed lag model with smoothness priors proposed by Shi Her (1973) is examined and an alternative ridge-type estimator is obtained utilizing restricted least squares approach, A Monte Carlo experiment is carried out to compare the behavior of these estimators.     

Empirical bayes estimation of functionals of unknown probability measures

The empirical Dayes approach to one and two sal-npie problcrns has beeir considered by Korwar and Hollander (1976), Holiander and Korwar (1976) and Phadia and Susarla (1979). In this article we essen- tially generalize their empirical Bayes results by replacing the inlicaro-functions of. the sets (?∞,x) and {X≦Y} by arbitrary mea5, irable functions h(x) and h(x,y). More speclfically, the ernpiricaion yes estimation of esrimabie paramerers of degree one ani KG,I;ti kliown probability measure Pon (R,R) is considered. The asymptotic optimality of the these estimators, obtaining the exact risk expressions, is established. Also the results of Dalal and Phad (1983) we extended to the estimation of an estimable parametric function of an unknow probability measure P on (R² , B²)     

Bayesian analysis of the difference of two proportions

In this article we give the expression of the prior distribution of p₁-P₂, where P₁ and P₂ and independent proportions with a beta prior each. The expression derived for the posterior distribution of P₁-P₂ then shows the closure of the beta-difference family for independent dual Bernoulli samples. Other bayesian results are also presented.     

Bayesian robustness of credible regions in the presence of nuisance parameters

The problem of finding the most robust γ-level credible region for the parameter of interest in the presence of a nuisance parameter, with respect to a class of ε-contaminated priors, is studied. The case of arbitrary con-taminations is first analyzed; it is proved that the most robust region for the parameter of interest is theγ-level highest marginal likelihood region (forγ ≥ 0.5). Then, the result is extended to any measurable (not necessarily one-to-one) function of the parameter. Finally, the case of contaminations assigning fixed probabilities to the sets of a partition of the parameter space is analyzed and a partial result is given.     

A bayesian method in determining the order of a finite state markov chain

In this paper, we use the Bayesian method in the application of hypothesis testing and model selection to determine the order of a Markov chain. The criteria used are based on Bayes factors with noninformative priors. Com¬parisons with the commonly used AIC and BIC criteria are made through an example and computer simulations. The results show that the proposed method is better than the AIC and BIC criteria, especially for Markov chains with higher orders and larger state spaces.