Page 480 - Probability and Statistical Inference

9. Confidence Interval Estimation 457

based on the sufficient statistics for θ. With n ≥ 2, let us denote
i

for i = 1, 2. Here, is the pooled sample variance.
Based on (4.5.8), we define the pivot

which has the Student’s t distribution with v = (n + n – 2) degrees of
v
2
1
freedom. Now, we have P{–t , < U < t v,α/2 } = 1 – α where t v,α/2 is the upper
v α/2
100(1 – ½α)% point of the Student’s t distribution with v degrees of free-
dom. Thus, we claim that

Now, writing , we have

as our (1 – α) two–sided confidence interval for µ – µ . !
1
2
Example 9.3.2 Difference of Negative Exponential Locations with a
Common Unknown Scale: Suppose that the random variables X , ..., X in
i1
are iid having the common pdf f(x; µ , σ), i = 1, 2, where we denote f(x; µ,
i
–1
σ) = σ exp{ – (x – µ)/σ}I(x > µ). Also let the X ’s be independent of the
1j
X ’s. We assume that all three parameters are unknown and α = (µ , µ , σ) ∈
2j
2
1
ℜ×ℜ×ℜ . With fixed α ∈ (0, 1), we wish to construct a (1 – α) two–sided
+
confidence interval for µ – µ (= κ(α)) based on the sufficient statistics for
2
1
α. With n ≥ 2, let us denote
for i = 1, 2.
Here, W is the pooled estimator of σ. It is easy to verify that W estimates
P
P
σ unbiasedly too. It is also easy to verify the following claims:

That is, V[W ] < V[W ], i = 1, 2 so that the pooled estimator W is indeed a
P
P
i
better unbiased estimator of s than either W , i = 1, 2.
i