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460 9. Confidence Interval Estimation

9.3.2 Comparing the Scale Parameters

The examples include estimation of the ratio of (i) the variances of two inde-
pendent normal populations, (ii) the scale parameters of two independent nega-
tive exponential populations, and (iii) the scale parameters of two independent
uniform populations.
Example 9.3.4 Ratio of Normal Variances: Recall the Example 4.5.3 as
needed. Suppose that the random variables X , ..., X are iid n ≥
i
ini
i1
2, i = 1, 2, and that the X ’s are independent of the X ’s. We assume that all
2j
1j
four parameters are unknown and (µ , σ ) ∈ ℜ × ℜ , i = 1, 2. With fixed α ∈
+
i
i
(0, 1), we wish to construct a (1 – α) two-sided confidence interval for
based on the sufficient statistics for θ(= (µ , µ , σ , σ )). Let us denote
1 2 1 2
for i = 1, 2 and consider the pivot
It should be clear that U is distributed as F n1–1,n2–1 since (n – 1) is
i
distributed as i = 1, 2, and these are also independent. As before, let us
denote the upper 100(α/2)% point of the F n1–1,n2–1 distribution by F n1–1,n2–1,α/
. See the Figure 9.3.2.
2

Figure 9.3.2. Area on the Right (or Left) of F n1–1,n2–1,α/2}
(or F ) Is α/2
n1–1mn2–1,1–α/2
Thus, we can write P{F n1–1,n2–1,1–α/2 < U F n1–1,n2–1,α/2 } = 1 – α and claim
that