Question

(17 points) Occasionally, a mean function in which the intercept is known to be 0 may be fit. Equivalently, our simple linear regression model is Y; = B1X: + € Assume in a fixed model (z values are fixed), this mean function is given by E[Y] =B120. (a) Assuming € are i.i.d following N(0,02), the residual sum of squares for this model, is SSE = +1(Y; – B12:) Show B1, the least squares estimate of B1. Show that ſi is unbiased and find the variance of $1. (b) Assuming are independently following N(0,of), we have a model with non-constant variance. If we fit the model under weighted least squares with some weight values wi, the weighted residual sum of squares for this model is WSSE = X-1 w:(Y; – B10:)2. Find Bwi that minimize the WSSE. Show that Bwi is unbiased. Under what condition can we make the variance of Bwl being ?

Answer 1

thanks and please rate the answer.

Ques: The SL R M ais Yi = Bikitei ein NIO 02) SSE 1 (a) 1s (4i-pixi)2 differentiate wirt prvend equate to zero in milli-pixi)(-2) = 0 Eyi ni aßi op 2 Eni ou 041 B Eyini estimate of fi E ui Thus, Å is least square I. You show unbiasedbens E [ @ 1 6 [ Eyimi] Eui² as ni's au non-stochastic so can be created as constants E ni ni Elyi) Ew2 J E ni B (Boxed) Eui² Bietis BI i So', 15 unbiased untimate off Varl & niye) = Eni Vanly) Eur) 2 Eur Eur 2 Var (Bil = 2 Σκα 11 Teni ; 22Z

tha

2) =O - 1 ७५॥ Oy Ewing I: Yo weighted deathe squares ws's 6 = Wil4i-Biri )? WSSE ² Wil4i-Bi Ki) (-21) api Ewiyini = Boje wille? fuir Ewi kiyi Yo nhaus undiewellness, rake pedalion El Boi Ewin? = E Wirli & W. Mi Elyor cyi) { Ewini² non-spachante {witiki {wilt? so, Boor is unbiased for ßi Var (Bean ni's are = Varel Ewistiy 1: Ewiri Elwini). +2 Cove Early Ewin? yo equate it to o2 condition TE Willi?) 2 equal 2/Ewin? Cov' (wi, wy)=0 Thus, the and {(Wiri)? required is wil wi2 = wilwirl) = 1 wito wi=1 = Ewini? - Wini 2 ou (Winie wil