Page 468 - Probability and Statistical Inference

9. Confidence Interval Estimation 445

Now, the equation (9.2.3) can be rewritten as

and thus, we can claim that is a 100(1 – α)% lower
confidence interval estimator for µ. !

The upper–sided a level test ⇒ 100(1 – α)% lower confidence
interval estimator (T (X), ∞) for θ.
L

The lower–sided a level test ⇒ 100(1 – α)% upper confidence
interval estimator (–∞, T (X)) for θ.
U
Example 9.2.2 Let X , ..., X be iid with the common exponential pdf θ –
1
n
1 exp{–x/θ}I(x > 0) with the unknown parameter θ ∈ ℜ . With preassigned α
+
∈ (0, 1), the UMP level α test for H : θ = θ versus H : θ > θ where θ is a
0
0
0
1
0
fixed positive real number, would be as follows:
where ,a is the upper 100α% point of the Chi–square distribution with
2n degrees of freedom. See the Figure 9.2.2.

Figure 9.2.2. The Shaded Area on the Right of , α Is a
The acceptance region (for H ) then corresponds to
0

Since this test has level a, we can write