Theory of Statistics - George Mason University

310 3 Basic Statistical Theory
For comparing two estimators, we may use the asymptotic relative efficiency.
The asymptotic relative efficiency of the estimators Sn and Tn for
g(θ) wrt P is defined as
ARE(Sn, Tn) = AMSE(Sn, g(θ), P)/AMSE(Tn, g(θ), P). (3.173)
The ARE is essentially a scalar concept; for vectors, we usually do one at
a time, ignoring covariances.
Asymptotic Significance
For use of asymptotic approximations for confidence sets and hypothesis testing,
we need a concept of asymptotic significance. As in the case of exact
significance, the concepts in confidence sets and hypothesis tests are essentially
the same.
We assume a family of distributions P, a sequence of statistics {Tn}, and
a sequence of tests {δ(Xn)} based on the iid random variables X1, . . ., Xn.
The test statistic δ(·) is defined in terms the decisions; it takes the value 1
for the case of deciding to reject H0 and conclude H1, and the value 0 for the
case of deciding not to reject H0.
Asymptotic Properties of Tests
In hypothesis testing, the standard setup is that we have an observable random
variable with a distribution in the family P. Our hypotheses concern a specific
member P ∈ P. We have a null hypothesis
and an alternative hypothesis
H0 : P ∈ P0
H1 : P ∈ P1,
where P0 ⊆ P, P1 ⊆ P, and P0 ∩ P1 = ∅.
Definition 3.19 (limiting size)
Letting β(δ(Xn), P) be the power function,
We define
β(δ(Xn), P) = PrP(δ(Xn) = 1).
if it exists, as the limiting size of the test.
Theory of Statistics c○2000–2013 James E. Gentle
lim
n→∞ sup β(δ(Xn), P), (3.174)
P ∈P0