Page 470 - Probability and Statistical Inference

9. Confidence Interval Estimation 447

The Location Case: With some a(θ), the distribution of
{T – a(θ)} would not involve θ for any θ ∈ Θ.

The Scale Case: With some b(θ), the distribution of
T/b(θ) would not involve θ for any θ ∈ Θ.

The Location–Scale Case: With some a(θ), b(θ), the
distribution of {T – a(θ)}/b(θ) would not involve θ for any θ ∈ Θ.

Definition 9.2.1 A pivot is a random variable U which functionally de-
pends on both the (minimal) sufficient statistic T and θθ θθ θ, but the distribution
of U does not involve θθ θθ θ for any θθ θθ θ ∈ Θ.
In the location, scale, and location–scale situations, when U, T and θ are
all real valued, the customary pivots are appropriate multiples of {T – a(θ)},
T/b(θ) or {T – a(θ)}/b(θ) respectively with suitable expressions of a(θ) and
b(θ).

We often demand that the distribution of the pivot must
coincide with one of the standard distributions so that a
standard statistical table can be utilized to determine the
appropriate percentiles of the distribution.

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Example 9.2.4 Let X be a random variable with its pdf f(x; θ) = θ exp{
–x/θ}I(x > 0) where θ(> 0) is the unknown parameter. Given some α ∈ (0,1),
we wish to construct a (1 – α) two–sided confidence interval for θ. The
statistic X is minimal sufficient for θ and the pdf of X belongs to the scale
family. The pdf of the pivot U = X/θ is given by g(u) = e I(u > 0). One can
–u
explicitly determine two positive numbers a < b such that P(U < a) = P(U >
b) = ½α so that P(a < U < b) = 1 – α. It can be easily checked that a = –log(1
– ½α) and b = –log(½α).
Since the distribution of U does not involve θ, we can
determine both a and b depending exclusively upon α.

Now observe that

which shows that J = (b X, a X) is a (1 – α) two–sided confidence interval
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estimator for θ. !