Theory of Statistics - George Mason University

294 3 Basic Statistical Theory
This is also called a (1 − α)100% confidence interval. The interval [TL, TU] is
not uniquely determined.
The concept extends easily to vector-valued parameters. Rather than taking
vectors TL and TU, however, we generally define an ellipsoidal region,
whose shape is determined by the covariances of the estimators.
A realization of the random interval, say [tL, tU], is also called a confidence
interval.
In practice, the interval is usually specified with respect to an estimator
of θ, T. If we know the sampling distribution of T − θ, we may determine c1
and c2 such that
Pr(c1 ≤ T − θ ≤ c2) = 1 − α;
and hence
Pr (T − c2 ≤ θ ≤ T − c1) = 1 − α.
If either TL or TU is infinite or corresponds to a bound on acceptable
values of θ, the confidence interval is one-sided. Suppose Θ = (a, b), where
a or b may be infinite. In equation (3.156), if TL = a, then TU is called an
upper confidence bound, and if TU = b, then TL is called a lower confidence
bound. (It is better not to use the terms “upper confidence interval” or “lower
confidence interval”, because of the possible ambiguity in these terms.)
For two-sided confidence intervals, we may seek to make the probability
on either side of T to be equal, to make c1 = −c2, and/or to minimize |c1| or
|c2|. This is similar in spirit to seeking an estimator with small variance.
We can use a pivot function g(T, θ) to form confidence intervals for the
parameter θ. We first form
Pr g(α/2) ≤ g(T, θ) ≤ g(1−α/2) = 1 − α,
where g(α/2) and g(1−α/2) are quantiles of the distribution of g(T, θ); that is,
Pr(g(T, θ) ≤ g(π)) = π.
If, as in the case considered above, g(T, θ) = T − θ, the resulting confidence
interval has the form
Pr T − g(1−α/2) ≤ θ ≤ T − g(α/2) = 1 − α.
Example 3.24 Confidence Interval for Mean of a Normal Distribution
Suppose Y1, Y2, . . ., Yn are iid as N(µ, σ2 ) distribution, and Y is the sample
mean. The quantity
n(n − 1)(Y − µ)
g(Y , µ) =
Yi
− Y 2 Theory of Statistics c○2000–2013 James E. Gentle