3
Due STA244
2/5/2005
Homework 2/12/2001 1. Recall from class that a non-central 2 (m, ) can be represented as a Poisson mixture of central 2 random variables, where Y P (/2) and X|Y 2 (m + 2y, 0). Find the mean and variance of a non-central Chi-squared random variable with m degrees of freedom and non-centrality parameter . 2. Consider the linear model Y = + N (0, 2 In ) where = X, X is n p rank r p and
(a) Show that if X is of full column rank r = p, then PX = X(X X)-1 X is an orthogonal projection onto the space spanned by the columns of X. (b) Find an expression for the projection PX in the non-full rank case (hint: use the Singular Value Decomposition Thm with X). (c) Show that QX I - PX is also an orthogonal projection on to the orthogonal complement of the span of X, S(X) . (d) Find the distribution of ||PX Y ||2 . (general rank case) (e) Find the distribution of ||QX Y ||2 . (general rank case) (f) Find the distribution of Y ||PX ||2 /||QX Y ||2 ). 3. Suppose we have a n p matrix Q and a p p upper triangular matrix R such that Q Q = Ip and QR = X. assume that X is of rank p (a) Show that R R = X X. ^ ^ (b) Show that = R-1 Q Y . Thus to compute , one computes z = Q Y then solves ^ the system of equations R = z by back substitution without explicit inversion of X X. (c) Show that QQ is an orthogonal projection of rank p onto the S(X) and that ^ Y = QQ Y . (d) Find e (the residuals) and residual sum of squares in terms of Y and Q. ^ (e) Show that the variance of a linear combination ...