November 8, 2011
1. Suppose App is a symmetric, idempotent, and positive denite matrix. Provide as much additional information about A as possible.
2. Suppose y is an n 1 random vector of responses, and X is an n p known design matrix.
November 8, 2007
1. Suppose y = X + , where has mean 0 and dispersion 2 I . Suppose L is estimable. Prove
that the ordinary least squares estimator of L is W X y if and only if X X W = L. (18 points)
2. Consider a completely randomized ex
September 27, 2011
1. Is it true that every symmetric matrix has at least one symmetric generalized inverse? Provide
a proof to support your answer.
2. Suppose S is a vector space in I n . Complete the following statement and prove that it
September 25, 2007
1. A matrix A : n n is said to be an orthogonal projector if A is both idempotent and symmetric.
Let y denote an arbitrary element of I n . Prove that if A is symmetric and idempotent, then Ay
is the closest point in th
November 6, 2012
1. Suppose y = X + , where y is a vector of responses, X is a known and xed n p
design matrix, is an unknown parameter vector in Rp , and is a random and unobserved
(a) What additional assumptions are necessar
November 11, 2009
Note: Points for each question are indicated in the left margin in square brackets. Simplify your results
to the greatest extent possible.
Consider the following model for a one-way classication with two concom
Homework #1 Solutions
1. (a) Part (i) of Corollary 1 to Lemma 1.4 follows from Lemma 1.4 and the corollary
to Lemma 1.1. Parts (ii) and (iii) are obtained by taking transposition on both
sides of the identities in part (i) and in
Homework #2 Solutions
1. (a) Yes! If 1 and 2 are both estimable, then there exist vectors a1 and a2 such
that E (a1 y ) = 1 and E (a2 y ) = 2 , in which case E (a1 y + a2 y ) = 1 + 2 .
(b) No! Suppose that is non-estimable. Then,
Homework #3 Solutions
1. (a) For the unconstrained model, the regression line is E (y ) = 0 + 2 x for x t,
and is E (y ) = (0 + 1 ) + (2 + 3 )x for x > t. This regression has a jump of
1 + 3 t at x = t, which may be undesirable.
September 25, 2012
1. Consider the general linear model discussed in class, where y = X + with E ( ) = 0.
(a) State the denition of a linearly estimable function c .
(b) State the denition of a least squares estimator of a linearly estimabl