Theory of Statistics - George Mason University

7.10 Asymptotic Confidence sets 547
if it exists.
Example (MS2). Suppose X1, . . ., X − n are iid from a distribution with
CDF PX and finite mean µ and variance σ 2 . Suppose σ 2 is known, and we
want to form a 1 −α level confidence interval for µ. Unless PX is specified, we
can only seek a confidence interval with asymptotic significance level 1−α. We
have an asymptotic pivot T(X, µ) = (X − µ)/σ, and √ nT has an asymptotic
N(0, 1) distribution. We then form an interval
C(X) = (C1(X), C2(X))
= (X − σz1−α/2/ √ :n, X + σz1−α/2/ √ :n),
where z1−α/2 = Φ −1 (1 −α/2) and Φ is the N(0, 1) CDF. Now consider Pr(µ ∈
C(X)). We have
Asymptotic Correctness and Accuracy
A confidence set C(X) for θ is 1 − α asymptotically correct if
lim n Pr(C(X) ∋ θ) = 1 − α.
A confidence set C(X) for θ is 1 −α l th -order asymptotically accurate if it
is 1 − α asymptotically correct and
lim n Pr(C(X) ∋ θ) = 1 − α + O(n −l/2 ).
Asymptotic Accuracy of Confidence sets
**************************
Constructing Asymptotic Confidence Sets
There are two straightforward ways of constructing good asymptotic confidence
sets.
One is based on an asymptotically pivotal function, that is one whose limiting
distribution does not depend on any parameters other than the one of
the confidence set.
Another method is to invert the acceptance region of a test. The properties
of the test carry over to the confidence set.
The likelihood yields statistics with good asymptotic properties (for testing
or for confidence sets).
See Example 7.24.
Woodruff’s interval
Theory of Statistics c○2000–2013 James E. Gentle