Question

may I ask for help with part a and b

Problem 1 (Chapter 4 Misspecification of Mean Function) [25 points]: The kinetic energy of y= Bo+Bx +e, e~N(0,a) an object (y) is related with its velocity (x) through 6 N(0,0). Suppose we fit the data {(x, y), i = 1,..., n} based on y a,+a,x + E, (Bo 1 X = and X2 [5 points] Show that E(a) = (X X)'X' X,B, with à = (a) (b) [14 points] Based on the result from part (a), show that E(à,)- p,+ and express E(a,) in terms of , x, x, n, Bo and B. .

Answer 1

Solution The kinetic energy of an object ly) is melated with it vectocity (s through y = Bot B,x+ e, en N(0,0%) il y = X2B ta - Grue model nx nx22x) nx where X - [gov Geo - Yo]! x | nx2 : onde (). ., -(0.6.com But, the data { (x,y;), iz sn} based on the model: Y = Lot dix+ E, ENN 10, 002) le mabix form, Y= xx + € , EN No lo az I] nx nx2 2x1 nx MAL where, 12). = ( 61,6... En)?

now, we fit data {(x, y) } i = 161)n on 4=X&t € to estimate d using bast square estimation y = X X + E 2x Int! nxi a = (x'xJ' x'y - nxi (By using ols method Cordinary least squares)) now, clã ) = (x'xJ' x'ELY) (: x is non stoohastic) aec@(x'x]' x'ely) — the true mode) in equation ③ we put €(4) from Lie from 0 , Y = X264 € (:6(C)=0 >) E(Y) = x2 B so, ec@) = (x'x5' x'ely) - le(a) = (x' x ' x'x28]

- E fres simplicity consider only 2 variables ; say x,& la *-[13).*[.] **.6:]] q x+x27 24, +X (x+ (els". vatns. Como lo com att - (x,+7)] -(x1+x₂)

consides [:] [:*:) Pasom 0,9 90 >0 port @ we get; [x+ 27) (2. Bot B, (x²+x²) - (x1+x2) [Bo (xitxe) + Br(x2+x})] - (Mitx) { [28+ Bo (x2+x2}]} +2{Bo [xithe) +B. (x?+*3) Bo [(x, + x2 + 2(x, t) Hence Edo) + Bi{(ki***) + (x + x>)(x1+)}