Math 3018

Homework 7

Due: Sunday, April 10

1. Recall in the setting of MANOVA the treatment SSP B = Σgl=1 nl(¯xl − ¯x)(¯xl − ¯x) > and n = Σgl=1 nl . Let 1m (resp. 0m) be the m dimensional vector with all entries equal to 1 (resp. 0).

(a) Let

A = diag(n1, ..., ng−1) − n/1 (n1, ..., ng−1) > (n1, ..., ng−1),

where diag(n1, ..., ng−1) is the diagonal matrix with entries n1, ..., ng−1. Show that A is positive definite.

(b) Let P = A−1/2 and define the (g − 1) × n matrix

Let Q = P(Z − n/1 (n1, ..., ng−1) T 1 Tn ) be an (g − 1) × n matrix. Show that the rows of Q are orthonormal, i.e. QQT = Ig−1.

(c) Recall the data matrix

such that all rows are mutually independent and xlj i.i.d. ∼ Np(µl , Σ), j = 1, ..., nl . Let Y = QX. Show that the columns of Y > are mutually independent with distribution Np(˜µl , Σ), l = 1, ..., g − 1, where ˜µl is the lth column of the matrix

(n1(µ1 − µ), ..., ng−1(µg−1 − µ))PT.

(d) Show that B = Y T Y . Hint: Use the Sherman–Morrison formula you proved in HW1 for A−1 and note that Σg l=1 nl(¯xl − ¯x) = 0.

(e) Under H0 : µ1 = · · · = µg = µ, conclude B ∼ Wp(g − 1, Σ).

2. Problems from the textbook by Johnson and Wichern: 6.23, 6.31, 7.6, 7.8, 7.25.