Exercise 2 In multiple linear regression, we can estimate ß as follows by least square method. -1 1 B = (X"x)-{XTy e RCP+1)x1, XE RN«(p+1), xły p . YERNx1 (2.1) The regression model can be derived as follows: ỹ = Xß = X(XTx)-1xły = Hy = = (2.2) And in the above equation, X(x+x)-1XT is specifically referred to as H, hat matrix. H = X(XTX)-2XT = (2.3)
2.2 = An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter. (E[@] = 0) SSE (y – ģ)(y - ģ) Show that MSE = ô 2 is unbiased estimator through using N-p-1 N-p-1 following properties (2.4) - (2.8) & results of above problem 2.1 (10 pts] = - If c is a scalar, and A is a nx n square matrix, (c ER, A E R**n) C= Tr(c), TrícA) = cer(A) (2.4) If A, B are nx n square matrix that have same dimension, (A, B e R**) Tr(A + B) = Tr(A) + Tr(B), Tr(A - B) = Tr(A) - Tr(B) = = (2.5) If A is a nxn square matrix, A E Rnxn E[Tr(A)] = Tr(E[A]) (2.6) nxm If A is a nx m matrix and B is a mx n matrix (A € Rum, B e Rmx) Tr(AB) = Tr(BA) (2.7) If x is random vector, Var[x] = E[xx"] – E[x]E[x]?, E[xx'] = Var[x] + E[x]E[x] = + (2.8)
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