  We present a general methodology for performing statistical inference on the
components of a real-valued matrix parameter for which rows and columns are
subject to order restrictions. The proposed estimation procedure is based on an
iterative algorithm developed by Dykstra and Robertson (1982) for simple order
restriction on rows and columns of a matrix. For any order restrictions on rows
and columns of a matrix, sufficient conditions are derived for the algorithm to
converge in a single application of row and column operations. The new
algorithm is applicable to a broad collection of order restrictions. In
practice, it is easy to design a study such that the sufficient conditions
derived in this paper are satisfied. For instance, the sufficient conditions
are satisfied in a balanced design. Using the estimation procedure developed in
this article, a bootstrap test for order restrictions on rows and columns of a
matrix is proposed. Computer simulations for ordinal data were performed to
compare the proposed test with some existing test procedures in terms of size
and power. The new methodology is illustrated by applying it to a set of
ordinal data obtained from a toxicological study.
