Suppose that y is an n times 1 observable random vector that

Suppose that y is an n times 1 observable random vector that follows the linear model y = X beta + , where beta is a p times 1 vector of unknown parameters, is an observable random vector whose distribution is N(Q, sigma2I), and sigma2 is an unknown positive parameter. Let p* = rank(X) and Px = X(X\'X)-X\'. Let A be an n times n symmetric and idempotent matrix of constants. Let r = rank(A). Find constants c1, c2, and c3 and an n times 1 vector of constants a such that C1+ a\' y + c2 y\' A y has a central x2 distribution with degrees of freedom c3.

Solution

Since A is symmetric and idempotent A²=A

Hence diagonal entries of A are either 0 or 1

If r = Rank (A)

r = no of non zero rows in matrix A

Hence there are c3 independent variables with sum following chi square distribution.

Df = c3