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522 11. Likehood Ratio and Other Tests

H : σ ≠ σ . Assume normality. From (11.3.21), we find the observed value
1
1
2
of the test statistic:

With α = .10, one has F = 5.0503 and F = .19801. Since F
5,5,.05 5,5,.95 calc
lies between the two numbers. 19801 and 5.0503, we conclude at 10% level
that the two stock prices were equally variable during the six days under
investigation. !

The problem of testing the equality of means of two independent
normal populations with unknown and unequal variances is hard.
It is referred to as the Behrens-Fisher problem. For some ideas
and references, look at both the Exercises 11.3.15 and 13.2.10.

11.4 Bivariate Normal Observations

We have discussed LR tests to check the equality of means of two indepen-
dent normal populations. In some situations, however, the two normal popu-
lations may be dependent. Recall the Example 9.3.3. Different test procedures
are used in practice in order to handle such problems.
Suppose that the pairs of random variables (X , X ) are iid bivariate nor-
2i
1i
mal, N (µ , µ , ρ), i = 1, ..., n(≥ 2). Here we assume that all five
1
2
2
parameters are unknown, (µ , σ ) ∈ ℜ × ℜ , l = 1, 2 and –1 < ρ < 1. Test
+
l
l
procedures are summarized for the population correlation coefficient ρ and
for comparing the means µ , µ as well as the variances .
1 2
11.4.1 Comparing the Means: The Paired Difference t Method
With fixed α ∈ (0, 1), we wish to find a level α test for a null hypothesis H :
0
µ = µ against the upper-, lower-, or two-sided alternative hypothesis H .
1
1
2
The methodology from the Section 11.3.1 will not apply here. Let us denote

Observe that Y , ..., Y are iid N(µ - µ , σ ) where σ = – 2ρσ σ .
2
2
1
2
n
1
1
2
2
Since the mean µ – µ and the variance σ of the common normal distribu-
2
1
tion of the Y’s are unknown, the two-sample problem on hand is reduced