14.381: Statistics
Fall, 2007
Midterm Examination
1. True, false, or uncertain? Explain your answer briefly.
(a) If θˆ1 and θˆ2 are unbiased estimators for parameter θ, then 31 θˆ1 + 23 θˆ2 must
be unbiased for θ.
(b) Estimators obtained by the maximum likelihood method are unbiased.
¯ ·5
(c) If X1 , X2 , . . . , Xn are i.i.d. normal random variables N (µ, σ 2 ), then X
P
¯ 2 /20, where X
¯ is the sample mean.
is independent of ni=1 (Xi − X)
2. Let us have two independent random samples: X1 , . . . , Xn is a sample from
N (µx , σx2 ), and Y1 , . . . , Ym is a sample from N (µy , σy2 ).
(a) Write down a joint pdf for {X1 , . . . , Xn , Y1 , . . . , Ym }.
(b) Find a 4-dimensional sufficient statistic.
(c) Find the MLE of σx2 and σy2 .
(d) Assume for this question only that σx2 = σy2 = σ 2 . Find the MLE for σ 2 .
(e) Find a LR test statistic for testing H0 : σx2 = σy2 .
(f) Suggest an exact test for testing H0 : σx2 = σy2 . Hint: if U ∼ x2p and V ∼ x2q
are independent, then
U/p
V /q
∼ Fp,q (Fisher distribution)
(g) Assume that n = 100, m = 50, s2x = 5, s2y = 6.
Test the hypothesis
H0 : σx2 = σy2 using the test received in (f) at 95% level.
Tables of
quantailes of Fisher distribution are provided.
6 σy2 by using an asymptotic LR test and
(h) Test H0 : σx2 = σy2 vs. H1 : σx2 =
the data as in (g).
(i) Suggest a confidence set for β =
1
σx2
.
σy2
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14.381 Statistical Method in Economics
Fall 2013
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