LINEAR MODEL (STATISTICS)

LINEAR MODEL (STATISTICS)

I ONLY NEED HELP WITH QUESTION 2.4 BECAUSE THE REST OF THE QUESTIONS HAVE SOLUTIONS

QUESTION 2 [22 Marks] Consider a linear model with 1 1 0 0 31 1 1 0 o 33 ,u _ 1 1 0 o _ 44 _ a1 X_1100'y_36\"8_ a2 1 0 1 0 38 a3 1 0 0 1 26 2.1 Write the normal equations. (6) 2.2 Find two generalized inverses G1 and G? of XX. (6) 2.3 Use your result in part (2.2) considering E = Gx'y in general, to nd two estimates 3 of ,6 . (5) .4 Find two linear combinations of the elements 3 that are the same for both estimates and two that are different. (5) Consider a linear model coith X = 51 33 BE O 6 2 2 O 36 O 0 O 26 2 . ) The normal equations xly = (x/ x ) B 1 0 0 33 O 0 2 2 O 0 1 3 6 38 26 208 1 u 14y 4 4 0 38 O 2 2 26 1 . 23 The above system reprents normal equations 2.2 Since XX) =0 and rank of * h = 3 Now Two find generalized inverdes , and 62 ofWe have obtain a none singular minor of ranks 3. "let Me . 4 O NOW MY. = O O 1 Hence 67 = O 1 1/ 4 10 O O O 1 Now again obtain a. non- singular minor of rank ? Jel M 2 = 0 114 O 4 0 O O . O - 171 -1 Heng Gaze - 1 -1 Hence G, and G12 all two generalized inverse of xox . 2.3 Now first estimate of B. BE GrixlyB. 2 O O 208 1 / 4 O quy 36 O O O 38 38 O O O 1 26 26 36 2 2 38 26 Noco Find estimate of Bo - BZ = Gzxly Yy O 208 9.5 O O 14 4 2 6 2 2 O - / 38 80 O O O 2 2 26 Hence Bland By are estimates of B corresponding to 67 , and 612