2. Show that, under MLR (1), (a) [4] H is symmetric and idempotent; (b) [4] rank(H) = p+1; (c) [4] (XB)(In — H)(Xß) = 0; - (d) [5] y¹(In — H)y/o² ~ X²/−p−1· - where ẞ(Bo ẞ1 Bp) and ... Yi = = ß³xi + €i, i = 1, . . ., n, Ꮖ ; = (1 xil ... xip) for i 1,. assumptions about & are satisfied, i.e., ε¿ IID = (1) ..., n. We assume that all the DD N(0, 0²). Suppose that n > p+1. We denote the design matrix for the MLR as X and the response vector as y. We assume that XTX is invertible and denote the hat matrix as H, i.e. H = X(X'X)¯¹X'.
2. Show that, under MLR (1), (a) [4] H is symmetric and idempotent; (b) [4] rank(H) = p+1; (c) [4] (XB)(In — H)(Xß) = 0; - (d) [5] y¹(In — H)y/o² ~ X²/−p−1· - where ẞ(Bo ẞ1 Bp) and ... Yi = = ß³xi + €i, i = 1, . . ., n, Ꮖ ; = (1 xil ... xip) for i 1,. assumptions about & are satisfied, i.e., ε¿ IID = (1) ..., n. We assume that all the DD N(0, 0²). Suppose that n > p+1. We denote the design matrix for the MLR as X and the response vector as y. We assume that XTX is invertible and denote the hat matrix as H, i.e. H = X(X'X)¯¹X'.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![2. Show that, under MLR (1),
(a) [4] H is symmetric and idempotent;
(b) [4] rank(H) = p+1;
(c) [4] (XB)(In — H)(Xß) = 0;
-
(d) [5] y¹(In — H)y/o² ~ X²/−p−1·
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76f5089c-8cf5-4463-b6fc-dd4c86631dd9%2Fcf6aa1c0-989b-45e6-9eaf-45b02c4066c9%2F2q2l9xr_processed.png&w=3840&q=75)
Transcribed Image Text:2. Show that, under MLR (1),
(a) [4] H is symmetric and idempotent;
(b) [4] rank(H) = p+1;
(c) [4] (XB)(In — H)(Xß) = 0;
-
(d) [5] y¹(In — H)y/o² ~ X²/−p−1·
-
Transcribed Image Text:where ẞ(Bo ẞ1 Bp) and
...
Yi =
= ß³xi + €i,
i = 1, . . ., n,
Ꮖ ;
=
(1 xil
...
xip) for i
1,.
assumptions about & are satisfied, i.e., ε¿
IID
=
(1)
..., n. We assume that all the
DD N(0, 0²). Suppose that n > p+1. We denote
the design matrix for the MLR as X and the response vector as y. We assume that XTX is
invertible and denote the hat matrix as H, i.e. H = X(X'X)¯¹X'.
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