Affiliation:
1. Department of Applied Mathematics , University of Colorado , Box 526 , Boulder CO 80309-0526 , USA
Abstract
Abstract
Order statistics arising from 𝑚 independent but not identically distributed random variables are typically constructed by arranging some
X
1
,
X
2
,
…
,
X
m
X_{1},X_{2},\ldots,X_{m}
, with
X
i
X_{i}
having distribution function
F
i
(
x
)
F_{i}(x)
, in increasing order denoted as
X
(
1
)
≤
X
(
2
)
≤
⋯
≤
X
(
m
)
X_{(1)}\leq X_{(2)}\leq\cdots\leq X_{(m)}
.
In this case,
X
(
i
)
X_{(i)}
is not necessarily associated with
F
i
(
x
)
F_{i}(x)
.
Assuming one can simulate values from each distribution, one can generate such “non-iid” order statistics by simulating
X
i
X_{i}
from
F
i
F_{i}
, for
i
=
1
,
2
,
…
,
m
i=1,2,\ldots,m
, and arranging them in order.
In this paper, we consider the problem of simulating ordered values
X
(
1
)
,
X
(
2
)
,
…
,
X
(
m
)
X_{(1)},X_{(2)},\ldots,X_{(m)}
such that the marginal distribution of
X
(
i
)
X_{(i)}
is
F
i
(
x
)
F_{i}(x)
.
This problem arises in Bayesian principal components analysis (BPCA) where the
X
i
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.
Subject
Applied Mathematics,Statistics and Probability