| 摘要: | Let X-1,...,X-p be independent random variables with densities f(i)(x(i)\theta(i)), i = 1,...,p. Suppose that theta(1),...,theta(p) are exchangeable random variables. In this paper we consider the Bayesian robustness or posterior sensitivity analysis with respect to the class of epsilon-contaminated prior distributions Gamma(AM)={(1-epsilon)pi(0)+epsilon q: q is an element of(LAM)}, where L-AM consists of all the exchangeable distributions on (theta(1),...,theta(p)) and for each distribution in L-AM which can be expressed as a mixture of product of i.i.d. distributions. Most of the previous studies on posterior sensitivity analysis focus on a one-dimensional parameter space, and the posterior quantities considered in those analyses are the posterior probability of a subset of the parameter space, the posterior mean, and the posterior variance. The loss function is usually not taken into consideration, or if it is, only the zero-one or squared error loss is considered. It is worthwhile to consider other loss functions and other interesting posterior quantities, which are the goals of the paper. Under the weighted squared error loss L(theta, delta)=Sigma(i=1)(p) w(i)(theta(i))(theta(i)-delta(i))(2), we study the posterior sensitivity of the Bayes action, the posterior expected loss and some interested posterior quantities with respect to the class Gamma(AM). By using the de Finetti mixture representation of exchangeable random variables, the problem of finding the range of each component of the Bayes action over the class Gamma(AM) can be reduced to a much smaller class of distributions with support at at most p points. The range of other posterior quantities can also be obtained in a similar way. One example for simultaneous estimation of independent Poisson means is given. In this example we compute the ranges of Bayes action and discuss their posterior robustness issues. (C) 1997 Elsevier Science B.V. |
