A Restricted Least Squares Problem* HEY WORDS Least Squares Regression Analysis

Let X = (zii) be a fixed m X n matrix of reals and Y = (yi) be a fixed n-dimensional column vector. The usual least squares problem is to minimize (1) L(X) = (Y- X'X)'(Y- X'X), where A is a m-dimensional column vector and AT denotes the transpose of the math A. In this note we consider minimizing L(X) where X is subject to the restriction Xi 2 0 for i = 1, 2, , m. In the literature various types of inequality restrictions have been considered in least squares and regression problems [3], and the usual solution seems to involve linear programming. Below we solve our stated problem by considering 2 " unrestricted problems. The real valued function L(X) can be considered defined on ([0, m))". Now so that, with Xi fixed for all j # k, we have either lirnk,-- L(X) = +m or L(X) constant. The latter situation occurs only when xik = 0 for all i = 1, 2, 9., m. Therefore, for the purpose of minimiz-ing L(X), we could restrict X to a closed and bounded subset of R " and thus there is a X e (IO, a)) " which minimizes L ( A). If we solve the system we find the solution to the unrestricted problem (by solving XX'X = XY, the usual normal equation). If this X satisfies Xi 2 0, i = l, 2,-. * , m, then no further work need be done. Otherwise, a