Urgent help is neede please 4. Let B be an n-dimensional random vector of zero mean and positive-definite covariance matrix Q. Suppose measurements of the formy = WB are made where the rank of W is m. If ß is the linear minimumvariance estimate of ß based on y, show that the covariance of theerror B - B has rank n - m.

Given that is an n-dimensional random vector of mean= 0 and positive definite covariance matrix..…

Answered: linear minimum ovariance of the

Elementary Linear Algebra (MindTap Course List)

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ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

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Let B be an n-dimensional random vector of zero mean and positive- definite covariance matrix Q. Suppose measurements of the form y = WB are made where the rank of W is m. If B is the linear minimum variance estimate of B based on y, show that the covariance of the error B - B has rank n - m.
4. Let X1, X2, X3, X4 denote the variables bill length, bill depth, flipper length and body mass, respectively. Suppose that we introduce two new variables: Y1 = 3X1 + 2X2 and Y2 = X2 + X3 + X4. Let Y be the dataset recording variables Y1 and Y2 of the same individuals as in X. Find the mean vector and covariance matrix of Y.
Suppose three tests are administered to a random sample of college students. Let X1, ..., XN be observation vectors in R3 that list the three scores of each student, and for j = 1, 2, 3, let xj denote a student's score on the j th exam. Suppose the covariance matrix of the data is 2 6. Let y be an “index" of student performance, with y = c,x1 + C2X2 + C3X3 and cỉ + c² + c = 1. Choose c1, C2, C3 %3D %3D so that the variance of y over the data set is as large as possible. [Hint: The eigenvalues of the sample covariance matrix are 1 = 3, 6, and 9.1
By hand, compute the correlation matrix R that goes with the covariance matrix
3. Consider the vector B1 B = B2 with variance-covariance matrix Var(B,) Var(ß) = ( Cov(B1, B2) Cov (β, β ) Var(8,) ) and the matrix 3 -1 A = 1 Show – for this example – that Var(AB) = AVar(B)A'.
4. Let X'= (X₁,..., Xn) be an n-dimensional random vector whose covariance matrix exists. Let A be an m x n matrix of constants. Then Cov(AX) = ACov(X)A'. True or false, give reasoning.
Suppose the x-coordinates of the data (x1, V1), .., (x,, Vn) are in mean deviation form, so that E x; = 0. Show that if X is the design matrix for the least-squares line in this case, then XTX is a diagonal matrix.
Consider the following two data vectors: A = [3,7,11]B=[3,2,1] Calculate the covariance matrix of these two vectors.
Can you show that Q is an intensity matrix?
Consider the statistical linear model Y = A0 + e with m = vations, a parameter vector 0 = (61,02, 03)" with n = 3 components and e ~ N(0; 200² I6) having mean 0 and component variance 200². A realisation of this model for data Y and design matrix A is given by 8 7 1 6 obser- €1 580.8 -140.2 2 3 1 €2 219.0 7 3 4 €3 02 Өз -64.5 6 3 5 €4 73.4 2 1 5 E5 403.8 5 7 3 €6 (a) Find the Maximum Likelihood (ML) estimate 0 of 0 and the co- variance matrix Cov(@) of the ML estimate ở (b) If 0 has prior distribution 0 ~ N(0, 10²I3), then find the distribu- tion of posterior (0|Y) using ridge regression. (c) Show that the component correlations Corr(0;|Y), (0;|Y)) of (0|Y) are smaller than those of Corr(0;, 0;) of the ML estimate 0 for i + j.
Express the model Y₁ = Bo + Bixi + B₁x² + €i i=1,2,..., 5 where the &; have mean zero, variance o² and are uncorrelated, as a general linear model in matrix form by specifying Y, X, 3 and e.
Estimation of the model resulted in the following least squares estimates and corresponding estimated covariance matrix by var (b)) cou (b, bz, b3) = cou (b, ba) Cou (bị , ba) cov (bị , b3 ) Cou (b, , b3) var (b3) 1 var (b2) COu (b) , b3) cov (b2 , b3) 1 4. Which of the following is correct? O The hypothesis Họ : B2 = 0 versus the alternative H¡ : B2 ± 0 is not rejected using a 5% significance level and the hypothesis Ho : B2 = 1 versus the alternative H1 : Bz 1 is rejected using a 5% significance level O The hypothesis Họ : Bz = 0 versus the alternativeH : 32 ± 0 is rejected using a 5% significance level and the hypothesis Ho : B2 =1 versus the alternative H¡ : B2 # 1 is not rejected using a 5% significance level O The hypothesis Họ : B2 = 0 versus the alternative Hj : 32 ± 0 is not rejected using a 5% significance level and the hypothesis Ho : B2 = 1 versus the alternative H1 : B2 # 1 is not rejected using a 5% significance level %3D None of the above The hypothesis Họ : B2 = 0…

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