MTH207M (2024-25, O DD S EMESTER )
P ROBLEM S ET 3
1. Consider the model E(y1 ) = β 1 + β 2 , E(y2 ) = β 1 − β 2 , E(y3 ) = β 1 + 2β 2 with the usual
assumptions. Obtain the BLUE of 2β 1 + β 2 and find its variance. Also, find β̂.
2. Consider the model E(y1 ) = 2β 1 + β 2 , E(y2 ) = β 1 − β 2 , E(y3 ) = β 1 + αβ 2 with the usual
assumptions. Determine α such that the BLUEs of β 1 , β 2 are uncorrelated.
3. Consider the model E(y1 ) = β 1 + β 2 , E(y2 ) = 2β 1 , E(y3 ) = β 1 − β 2 with the usual assump-
tions. Find the RSS.
4. Consider the one-way Anova model:
yij = µ + αi + ϵij , i = 1, . . . , k, j = 1, . . . , ni ,
where ϵij are independent with the mean 0 and variance σ2 . What are the estimable
functions? Is the grand mean ȳ an unbiased estimator of µ?
5. Consider the model E(y1 ) = β 1 + 2β 2 , E(y2 ) = 2β 1 , E(y3 ) = β 1 + β 2 with the usual
assumptions. Find the RSS subject to the restriction β 1 = β 2 .
6. Let x1 , . . . , xn be real numbers with mean x̄. Consider the linear model
yi = α + β( xi − x̄ )
with the usual assumptions. Show that the BLUEs of α and β are uncorrelated.
7. Suppose for a linear model, there is no linear function c T y such that E(c T y) = 0. Further,
suppose for l T β, E(d T y) = l T β. What can we say about the BLUE of l T β?
1
8. Consider the standard linear model with usual assumptions. Let S = {c ∈ Rn | E(c T y) =
0}. Show that
(a) S is a subspace, [all functions c T y with c ∈ S are called the error functions]
(b) if d(S) = 1 and {d} is a basis of S, then for any p T β the BLUE is
cov(u T y, d T y) T
uT y − d y
var(d T y)
where E(u T y) = p T β.
9. Prove or disprove: Consider a standard linear model with usual assumptions. The for
any estimable linear function l T β, every unbiased estimator is of the form l T Gy for some
g-inverse G of X.
