Page 471 - Probability and Statistical Inference

448 9. Confidence Interval Estimation

Example 9.2.5 Let X , ..., X be iid Uniform(0, θ) where θ(> 0) is the
1
n
unknown parameter. Given some α ∈ (0,1), we wish to construct a (1 – α)
two-sided confidence interval for θ. The statistic T ≡ X n:n, the largest order
statistic, is minimal sufficient for θ and the pdf of T belongs to the scale
n–1
family. The pdf of the pivot U = T/θ is given by g(u) = nu I(0 < u <1). One
can explicitly determine two numbers 0 < a < b < 1 such that P(U < a) = P(U
> b) = ½α so that P(a < U < b) = 1 – α. It can be easily checked that a =
1/n
1/n
(½α) and b = (1 – ½α) . Observe that
–1
–1
which shows that J = (b X n:n, a X ) is a (1 – α) two–sided confidence
n:n
interval estimator for θ. !
Example 9.2.6 Negative Exponential Location Parameter with Known
Scale: Let X , ..., X be iid with the common negative exponential pdf f(x; θ)
1
n
= σ exp{–(x – θ)/σ}I(x > θ). Here, θ ∈ ℜ is the unknown parameter and we
–1
+
assume that σ ∈ ℜ is known. Given some α ∈ (0,1), we wish to construct a
(1 – α) two–sided confidence interval for θ. The statistic T ≡ X , the small-
n:1
est order statistic, is minimal sufficient for θ and the pdf of T belongs to the
–u
location family. The pdf of the pivot U = n(T – θ)/σ is given by g(u) = e I(u
> 0). One can explicitly determine a positive number b such that P(U > b) =
a so that we will then have P(0 < U < b) = 1 – α. It can be easily checked that
b = –log(α). Observe that

–1
which shows that J = (X – bn σ, X ) is a (1 – α) two-sided confidence
n:1
n:1
interval estimator for θ.

Figure 9.2.4. Standard Normal PDF: The Area on the
Right (or Left) of z (or – z ) Is α/2
α/2 α/2
Example 9.2.7 Normal Mean with Known Variance: Let X , ...,
1
2
X be iid N(µ, σ ) with the unknown parameter µ ∈ ℜ. We assume that
n