Theory of Statistics - George Mason University

440 5 Unbiased Point Estimation
and
MSA = n
MSE =
m
m
(Xi − X) 2 /(m − 1),
i=1
i=1 j=1
n
(Xij − Xi) 2 /(m(n − 1)).
Express MSA and MSE as quadratic forms using parts of Helmert matrices
and use Chochran’s theorem to show that they are independent.
5.8. Show that the quantities in expressions (5.95) and (5.96) have the chisquared
distributions claimed.
5.9. Show that the UMVUE of σ 2 , SSE/(m(n − 1)), given in Example 5.28
is the same as the UMVUE of σ 2 for the general linear model given in
equation (5.77).
Hint: Write the model given in equation (5.88) in the form of the general
linear model in equation (5.67).
5.10. Suppose Xij iid ∼ N(µi, σ 2 ) for i = 1, . . ., m and j = 1, . . ., n. (Compare the
one-way AOV model of Examples 5.28, 5.29, and 5.30.)
a) Determine the UMVUE Tmn(X) of σ 2 .
b) Show that Tmn(X) is consistent in mean squared error for σ 2 as m →
∞ and n remains fixed.
c) Show that Tmn(X) is consistent in mean squared error for σ 2 as n →
∞ and m remains fixed.
5.11. Show that the sample variance S 2 is the UMVUE of σ 2 in equation (5.105)
of Example 5.32. Hence, determine the UMVUE of V(Y ).
5.12. Show that the variance of the Horvitz-Thompson estimator is as shown in
equation (5.107), for given πi and πij. This is tedious, but it requires very
little other than “advanced arithmetic” and simple properties of variances
of sums.
Theory of Statistics c○2000–2013 James E. Gentle