Page 485 - Probability and Statistical Inference

462 9. Confidence Interval Estimation

This leads to the following conclusion:

is a (1 – α) two–sided confidence interval estimator for σ /σ . !
1 2
Example 9.3.6 Ratio of Uniform Scales: Suppose that the random vari-
ables X , ..., X are iid Uniform(0, θ ), i = 1, 2. Also let the X ’s be indepen-
in
i1
i
1j
dent of the X ’s. We assume that both the parameters are unknown and θ =
2j
(θ , θ ) ∈ ℜ × ℜ . With fixed α ∈ (0, 1), we wish to construct a (1 – α) two-
+
+
1
2
sided confidence interval for θ /θ based on the sufficient statistics for (θ ,
2
1
1
θ ). Let us denote X for i = 1, 2 and we consider the
ij
2
following pivot:
It should be clear that the distribution function of is simply t for 0 <
n
t < 1 and zero otherwise, i = 1, 2. Thus, one has
distributed as iid standard exponential random variable. Recall the Example
4.2.5. We use the Exercise 4.3.4, part (ii) and claim that

has the pdf - e –|w| I(w ∈ ℜ). Then, we proceed to solve for a(> 0) such that
1
2
P(|– nlog(U)| < a} = 1 – α. In other words, we need
which leads to the expression a = log(1/α). Then, we can claim that
Hence, one has:

which leads to the following conclusion:

is a (1 – α) two–sided confidence interval estimator for the ratio θ /θ . Look
2
1
at the closely related Exercise 9.3.11. !