![Let Y be a linear function of X, i.e. Y = bo + bịX where bo and bl are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the values of bo and bi that minimizes Q (b) Use(a) to show that the minimal value of Q is Co - Hint: You may use the fact that Q(bg, b) [(Y-Y*)2] = Var (Y-Y*) +E(Y-Y*)]2 where Y*-bg + bİX and bi, bi are the values you get from (a).](http://img.homeworklib.com/questions/b291fde0-73ac-11ea-a64f-23938bd2de09.png?x-oss-process=image/resize,w_560)
Homework Answers
(2) Let Y be a linear function of X, i.e. Y- bo biX where bo and...
(2) Let Y be a linear function of X, i.e. Y- bo biX where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity we=E[rMb, that minimize«Q (a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ-ar 2 Cov2(x,Y) m Hint: You may use the fact that Q(bo,Y-YVar (Y -Y)+E (Y -Y)where Y.-bg + bİX...
(2) Let be a linear function of X, ie. = bo +b1X where bo and bi...
(2) Let be a linear function of X, ie. = bo +b1X where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ -c 3xy 2 Cov2 (X.y Hint: You may use the fat that (b,bE[(Y -Yar (Y - Y)E(Y - Y) where Y.-bg +...
I am having trouble with part b. Please explain. (2) Let Y be a linear function...
I am having trouble with part b.
Please explain.
(2) Let Y be a linear function of X, ie. Y lo +biX where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the valucs of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is 2 Cov2(x,y) Hint: You may use the fact that Q(bg, bị) E [(Y-Y*)2-Var...
1. Let X and Y b e random variables, with μΧ = E(X), μΥ = E(Y),...
1. Let X and Y b e random variables, with μΧ = E(X), μΥ = E(Y), σ炙= Var(X) and σ Var(Y) (2) Let Ỹ be a linear function of X, ie. Ỹ = +51X where bo and bl are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ....
6. This problem considers the simple linear regression model, that is, a model with a single...
6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable....
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,......
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,... ,n. Let b1 = s^y/8r and bo = Y - b1 t be the least squared estimators of B1 and Bo, respectively. We showed in class, that N(B; 02/) Y~N(BoB1 T;o2/n) and bi ~ are uncorrelated, i.e. o{Y;b} We also showed in class that bi and Y 0. = (a) Show that bo is...
Problem 7. Consider the simple linear regression model Y1 = Bo + BiX; +€; for i=1,2,...,n...
Problem 7. Consider the simple linear regression model Y1 = Bo + BiX; +€; for i=1,2,...,n where the errors Eį are uncorrelated, have mean zero and common variance Varſei] = 02. Suppose that the Xį are in centimeters and we want to write the model in inches. If one centimeter = c inch with c known, we can write the above model as Yį = y +71 Zitki where Zi is Xi converted to inches. Can you obtain the least-squared...
PART B: Application 5. Suppose that you observe a random variable X. and then, on the...
PART B: Application 5. Suppose that you observe a random variable X. and then, on the basis of the observed value. you attempt to predict the value of a second random variable Y. Let Y denote the predictor or an estimator of Y ; that is, if X is observed to equal , then Y is your prediction for the value of Y, and your goal is to choose Y so that it tends to be close to Y First,...
2. The linear regression model in matrix format is Y Χβ + e, with the usual definitions Let E(elX...
2. The linear regression model in matrix format is Y Χβ + e, with the usual definitions Let E(elX) 0 and T1 0 0 01 0 r2 00 0 0 0 0.0 0 γΝ 0 00 Notice that as a covariance matrix, Σ is bymmetric and nonnegative definite () Derive Var (0LS|x). (ii) Let B- CY be any other linear unbiased estimator where C' is an N x K function of X. Prove Var (BIX) 2 (X-x)-1 3. An oracle...
Consider the simple linear regression model y - e, where the errors €1, ,en are iid....
Consider the simple linear regression model y - e, where the errors €1, ,en are iid. random variables with Eki-0, var(G)-σ2, i-1, .. . ,n. Solve either one of the questions below. 1. Let Bi be the least squares estimator for B. Show that B is the best linear unbiased estimator for B1. (Note: you can read the proof in wikipedia, but you cannot use the matrix notation in this proof.) 2. Consider a new loss function Lx(A,%) 71 where...
(2) Let Y be a linear function of X, i.e. Y- bo biX where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity we=E[rMb, that minimize«Q (a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ-ar 2 Cov2(x,Y) m Hint: You may use the fact that Q(bo,Y-YVar (Y -Y)+E (Y -Y)where Y.-bg + bİX...
(2) Let be a linear function of X, ie. = bo +b1X where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ -c 3xy 2 Cov2 (X.y Hint: You may use the fat that (b,bE[(Y -Yar (Y - Y)E(Y - Y) where Y.-bg +...
I am having trouble with part b.
Please explain.
(2) Let Y be a linear function of X, ie. Y lo +biX where bo and bi are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the valucs of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is 2 Cov2(x,y) Hint: You may use the fact that Q(bg, bị) E [(Y-Y*)2-Var...
1. Let X and Y b e random variables, with μΧ = E(X), μΥ = E(Y), σ炙= Var(X) and σ Var(Y) (2) Let Ỹ be a linear function of X, ie. Ỹ = +51X where bo and bl are fixed real numbers. We want to minimize the discrepancy of Y from Y, i.e. minimizing the quantity (a) Find the values of bo and bi that minimizes Q (b) Use (a) to show that the minimal value of Q is σ....
6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable....
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,... ,n. Let b1 = s^y/8r and bo = Y - b1 t be the least squared estimators of B1 and Bo, respectively. We showed in class, that N(B; 02/) Y~N(BoB1 T;o2/n) and bi ~ are uncorrelated, i.e. o{Y;b} We also showed in class that bi and Y 0. = (a) Show that bo is...
Problem 7. Consider the simple linear regression model Y1 = Bo + BiX; +€; for i=1,2,...,n where the errors Eį are uncorrelated, have mean zero and common variance Varſei] = 02. Suppose that the Xį are in centimeters and we want to write the model in inches. If one centimeter = c inch with c known, we can write the above model as Yį = y +71 Zitki where Zi is Xi converted to inches. Can you obtain the least-squared...
PART B: Application 5. Suppose that you observe a random variable X. and then, on the basis of the observed value. you attempt to predict the value of a second random variable Y. Let Y denote the predictor or an estimator of Y ; that is, if X is observed to equal , then Y is your prediction for the value of Y, and your goal is to choose Y so that it tends to be close to Y First,...
2. The linear regression model in matrix format is Y Χβ + e, with the usual definitions Let E(elX) 0 and T1 0 0 01 0 r2 00 0 0 0 0.0 0 γΝ 0 00 Notice that as a covariance matrix, Σ is bymmetric and nonnegative definite () Derive Var (0LS|x). (ii) Let B- CY be any other linear unbiased estimator where C' is an N x K function of X. Prove Var (BIX) 2 (X-x)-1 3. An oracle...
Consider the simple linear regression model y - e, where the errors €1, ,en are iid. random variables with Eki-0, var(G)-σ2, i-1, .. . ,n. Solve either one of the questions below. 1. Let Bi be the least squares estimator for B. Show that B is the best linear unbiased estimator for B1. (Note: you can read the proof in wikipedia, but you cannot use the matrix notation in this proof.) 2. Consider a new loss function Lx(A,%) 71 where...
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