A pseudo restricted maximum likelihood estimator under multivariate simple tree order restriction and an algorithm

The minimum distance projection of a given matrix X \in R^pxq onto the order restricted cone in an appropriately defined inner product system, π(X|C_pxq), plays an important role in order restricted statistical inference since in many cases the restricted maximum likelihood estimator (RMLE) for a parameter matrix under an order restriction is the projection of the maximum likelihood estimator (MLE) without any restrictions onto the order restricted cone. The RMLE plays an important part in the maximum likelihood ratio tests. The computation for π(X|_pxq), however is currently a great challenge to researchers. It is known that the order relation $\preceq$ in R^p is a multivariate order relation if and only if it is generated from a closed convex cone C \in R^p, called an order generating cone. The collection of all matrices μ = (μ₁,...,μ_q) \in R^pxq whose columns satisfy the multivariate order restriction μ i \preceq μ i for all $(i, j)$ in a specified set $H \subset$ {1,...,q} x {1,...,q} is a closed convex cone C_pxq in R^pxq called an order restricted cone. For C_pxq created by multivariate simpletree order restriction and a given matrix X \in R^pxq, in this dissertation, a closed convex subset D(X)_pxq \subset C_pxq is defined. The projection of X onto this subset, π(X|D(X0_pxq), is studied. In addition, an algorithm for computing π(X|D(X)_pxq) is proposed and proved. The proposed algorithm for π(X|D(X)_pxq) only depends on projections of vectors onto the order generating cone. Thus, it converts the relatively difficult matrix projection problem to a much easier vector projection problems. It is also revealed that when q = 2, π(X|D(X)_pxq) = π(X|C_pxq) and if X \in C_pxq, π(X|D(X)_pxq) = π(X|C_pxq). With all these good properties we could treat the projection onto D(X)_pxq as the approximation of the projection onto $C_{pxq}.

Author

• Asfha, Huruy Debessay

Identifier

hdl:10057/21734

OAI identifier oai:identifier

oai:soar.wichita.edu:10057/21734