Controlled accuracy Gibbs sampling of order-constrained non-iid ordered random variates.

Order statistics arising from 푚 independent but not identically distributed random variables are typically constructed by arranging some X 1 , X 2 , ... , X m , with X i having distribution function F i ⁢ (x) , in increasing order denoted as X (1) ≤ X (2) ≤ ⋯ ≤ X (m) . In this case, X (i) is not necessarily associated with F i ⁢ (x) . Assuming one can simulate values from each distribution, one can generate such "non-iid" order statistics by simulating X i from F i , for i = 1 , 2 , ... , m , and arranging them in order. In this paper, we consider the problem of simulating ordered values X (1) , X (2) , ... , X (m) such that the marginal distribution of X (i) is F i ⁢ (x) . This problem arises in Bayesian principal components analysis (BPCA) where the X i are ordered eigenvalues that are a posteriori independent but not identically distributed. We propose a novel coupling-from-the-past algorithm to "perfectly" (up to computable order of accuracy) simulate such order-constrained non-iid order statistics. We demonstrate the effectiveness of our approach for several examples, including the BPCA problem. [ABSTRACT FROM AUTHOR]