Page 484 - Probability and Statistical Inference

9. Confidence Interval Estimation 461

Hence, we conclude that

is a (1 – α) two-sided confidence interval estimator for the variance ratio
By taking the square root throughout, it follows immediately from
(9.3.11) that

is a (1 – α) two–sided confidence interval estimator for the ratio σ /σ . !
1 2

Example 9.3.5 Ratio of Negative Exponential Scales: Suppose that
the random variables X , ..., X are iid having the common pdf f(x; µ ,
ini
i1
i
–1
s ), n ≥ 2, i = 1, 2, where f(x; µ, σ) = σ exp{–(x – µ)/σ}I(x > µ). Also
i
i
let the X ’s be independent of the X ’s. Here we assume that all four
2j
1j
+
parameters are unknown and (µ , σ ) ∈ ℜ × ℜ , i = 1, 2. With fixed α ∈ (0,
i
i
1), we wish to construct a (1 – α) two–sided confidence interval for σ /
1
σ based on the sufficient statistics for θ ( = (µ , µ , σ , σ )). We denote
2 1 2 1 2
for i = 1, 2 and consider the pivot
It is clear that U is distributed as F 2n1–2,2n2–2 since 2(n – 1)W /σ is distrib-
l
l
i
uted as and these are also independent. As before, let us
denote the upper 100(α/2)% point of the F 2n –2,2n – 2 distribution by
2
1
F n –2,2n –2,α/2 . Thus, we can write P{F 2n –2,2n –2,1–α/2 < U < F 2n –2,2n –2,α/2 } =
2
1
1 – α and hence claim that 1 2 1 2