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

Abstract

The minimum distance projection of a given matrix $X \in R^{pxq}$ onto the order restricted cone in an appropriately defined inner product system, $\pi(X|C_{pxq}),$ 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 $\pi(X|_{pxq}),$ 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 $\mu = (\mu_1,...,\mu_q) \in R^{pxq}$ whose columns satisfy the multivariate order restriction $\mu i \preceq \mu i$ 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, $\pi(X|D(X0_{pxq})$, is studied. In addition, an algorithm for computing $\pi(X|D(X)_{pxq})$ is proposed and proved. The proposed algorithm for $\pi(X|D(X)_{pxq})$ 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, $\pi(X|D(X)_{pxq}) = \pi(X|C_{pxq})$ and if $X \in C_{pxq}$, $\pi(X|D(X)_{pxq}) = \pi(X|C_{pxq})$. With all these good properties we could treat the projection onto $D(X)_{pxq}$ as the approximation of the projection onto $C_{pxq}.