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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
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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: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'.
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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