1. This is a more general version of Problem C.1. Let Y 1, Y 2,c, Yn be n pairwise uncorrelated random variables with common mean m and common variance s2. Let Y denote the sample average. (i) Define...

1. This is a more general version of Problem C.1. Let
Y1,
Y2,c,
Yn
be
n
pairwise uncorrelated random variables with common mean
m
and common variance s2. Let
Y
denote the sample average.

(i) Define the class of
linear estimators
of m by
Wa
5
a1Y1 1
a2Y2 1 c1
anYn, where the
ai
are constants. What restriction on the
ai
is needed for
Wa
to be an unbiased estimator of m?

(ii) Find Var 1Wa
2.

(iii) For any numbers
a1,
a2,c,
an, the following inequality holds: (a1 1
a2 1 p 1
an
22/n
#
a21 1
a22 1 p 1
a2n
. Use this, along with parts (i) and (ii), to show that Var 1Wa
2 $ Var(Y) whenever
Wa
is unbiased, so that
Y
is the
best linear unbiased estimator. [Hint: What does the inequality become when the
ai
satisfy the restriction from part (i)?]