3. In the usual setup for multiple regression, e, hi, Vo) are the raw residual, leverage, and the...
Considering multiple linear regression models, we compute the regression of Y, an n x 1 vector, on an n x (p+1) full rank matrix X. As usual, H = X(XT X)-1 XT is the hat matrix with elements hij at the ith row and jth column. The residual is e; = yi - Ýi. (a) (7 points) Let Y be an n x 1 vector with 1 as its first element and Os elsewhere. Show that the elements of the...
Considering multiple linear regression models, we compute the regression of Y, an n x 1 vector, on an n x (p+1) full rank matrix X. As usual, H = X(XT X)-1 XT is the hat matrix with elements hij at the ith row and jth column. The residual is e; = yi - Ýi. (a) (7 points) Let Y be an n x 1 vector with 1 as its first element and Os elsewhere. Show that the elements of the...
linear statistics modeling and regression 2) Suppose you have multiple regression set up Ynxi XnxpBpxi Sxl and f ~ N(0nx1, σ21.). P Po X(X,X)-X, be the projection matrix on the column space of X. a) Show residual vector, e = (1,-P)Y. Here e is the vector of residuals ei S. b) Show that the variance of e, is 1 - Pi, where P is the i, j th entry of the matrix P c) Show that the sample covariance of...
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b) Write the matrix form of the model. Label the response vector, design matrix, coefficient vector, and error vector, and specify the dimensions and elements for each. (c) Write the likelihood, log-likelihood, and in matrix form. aB (d) Solve : 0 for β, the MLE...
Please solve the problems (d),(e),(f),(g) 3. (20 pts) Do not use a computer to complete the answers for this problem. Show your work in all problems. Consider the following data: 2 -2 2 1 0 4 0 2 10 0 -2 8 Consider the multiple regression model where ci ~ iid N(0, σ*) for i 1, 2, 3, 4, 5. (a) Fill in the values for response vector, design matrix and coefficient vector: X- (b) Find the vector of least...
linear stat modeling & regression 1) Consider n data points with 3 covariates and observations {xn, ^i2, xi3,yid; i,,n, and you fit the following model, y Bi+Br2+Br+e that is yi A) +Ari,1 +Ari,2 +Buri,3 + єї where є,'s are independent normal distribution with mean zero and variance ơ2 . H the vectors of (Y1, . . . ,Yn). Assume the covariates are centered: Σίχί,,-0, k = 1,2,3. ere, n = 50, Let L are Assume, X'X is a diagonal matrix...