Question

Чuauranc lепп руло F. In Section 10.2, show that: 1. E (lack of fit mean square) is as given. 2. E (residual mean square) = o? if the model is correct, that is, if B2 = 0. (Use the result in Section 10.3.) = Bo + BX, but the model El G. Suppose we fit, by least squares, the model El Be + B.X + B/X is actually the true response function, for X 21. If we use five obser of Y at X-values X = 5, 8, 10, 20, 40 to estimate B, and B, in the model actually fitted biases will be introduced into the estimates? model E(Y) = ve use five observations

G only

Answer 1

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True response function & Ely) = Bot Bi X + B2 False function : E(4)= Be + Bix Estinate of so, ßi from false function. . Yi= Bot Bixi + Ei izl...n. OLSE of Bo and Bi. 5 - Že2 - .(Yi - Bo – paxi)? as =0 80 = ý ßi as 0 - 81 = 2 (xi-x) (ys - y) 2 (xi-x)2 ß = 2 Cxi - ze) yi - S (xi - Z) 9 {(xi-2) 2 11 Ā , (xi-) yi S. (x-7) Bias Elpi - Bi) = 2 (xi-x) Elyi) - Bi (xi-sje 22 (xi _x) (Bo + B1 * xi +82) (xine) 2 + Bloxit aut - Bi S. using Expection I ofy of true model?

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Elfira 2 (xi - ze)" Bo + (xi-x)ari Bi + (x-7), 82 "El - 2 (xi-x² Bi + 2 (1-x) $2. {(xi-ze)? & Bi + (n - x 2 1 2 - Ž (xi-E) = n - 2 3 4 al Note = 2 HM = n ) { t ki ni Bias in estimate of Bi EC Bi - Bi) = n-ion HM o - 2bu-22 ß2 E (Ro) = Ely) - E(81) 2 = [Bot Bił + $2 2 2 - [ Bit nan prja य = Bo + Brx +32 2 - Brt - (n. In you ) B2 X 'n 121 x & Cri-zje 322

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Bias in estimate of Bo Elpo- Bo) = ß2 18 t - (1 - 1 2 3 - E (xi-zeja 2 (xi-z)? / X=(5,8,10,20,40). { aci = 83 n-5 n=5 ă = 16.6 =10 HM = n 5 0.5 1+1+1+1+1 15 8 10 20 40 f (x==3* - (+ 8.6+66+4 84 123 4? = 8112 B2 (-0.00406). L J - Bias CB1) = 82 5 - 5x16.6 10 811-2 Beas (81) = -0.004182 Bias (60) = / - (83-83). $11.2 a foul - 24.1.) B2 - (0.1 -0.0921) B2 Bias (86) = 0.0079 82