Prove that the covariance matrix of a random vector if

Question

1. Let Y =  be a random vector with mean vector and covariance matrix

(a) Let Z = 2Y₁ + 3Y₂ + Y₃. Find E[Z] and Var[Z].

(b) Z₁ = Y₁ + Y₂ + Y₃ and Z₂ = Y₁ + Y₂ - 2Y₃. Find E[Z] and Cov[Z], where Z = [Z₁Z₂]'.

(c) Let Z = Y₁² + Y₂² + 2Y₁Y₃ + Y₃². Find E[Z].

2. Prove that the covariance matrix of a random vector, if exists, is a non-negative definite matrix.

3. For fixed matrices A and B,

Cov(AY , BW) = ACov(Y , W)B'.

4. For the general linear model, prove that the predicted value Yˆ is invariant under a full rank linear transformation on the X.

5. Consider the Two-Factor ANOVA model discussed in class. Assume there are i = 1, . . . , 3 levels of the first factor, j = 1, . . . , 2 levels of the second factor, and k = 1, . . . 3 response for each treatment combination.

a. Write an appropriate general linear model equation in the form Y = Xβ + ∈ by giving each term Y , X, β, and ∈ explicitly.

b. Give the expected response vector E[ Y ] in terms of your model parameters.

c. Calculate the rank of the design matrix X. (Use R for this if you prefer.)

d. Calculate X'X and find the rank of the resulting matrix. (Use R for this if you prefer.)

e. Show that X'X is singular.

f. Find a generalized inverse for X'X. (HINT: Find an invertible submatrix of X'X with dimension equal to its rank.)

g. Give the form of all Least Squares Estimators for β, in terms of the components of the arbitrary vector h.

h. Find a basis of linear functions that describes all estimable functions for this problem.

i. Determine whether the following linear functions are estimable.

i. µ

ii. µ + α₁ - β₂

iii. µ + α₂ + β₂ + (α_β)₂₂

iv.( αβ)₁₂ - (αβ)₃₁

j. For each function of part i that is estimable, find the Best Linear Unbiased Estimator (BLUE) for that function.

6. Let Y_i,.........Y₆ denote the yield of a production process on six consecutive days. Machine A was on day 1,3 and 5, while machine B was used on days 2, 4and 6 consider the models:

(i) Y_i = β₀ +(- 1)^i β₁+ε

(II) Y_i = β₀ +(- 1)^i β₁ + iβ₂ + ε_i

In each case, assume that E(ε_i) = 0 and var(ε_i) = σ₂. obtain the least squares estimate of under each model, and show that the ratio of Var( βˆ1) under model (i) to Var( βˆ2) under model ( ii ) is 32/35.