Theory of Statistics - George Mason University

524 7 Statistical Hypotheses and Confidence Sets
Definition 7.2 (Chernoff consistency)
The sequence of tests {δn} with power function β(δ(Xn), P) is Chernoffconsistent
for the test iff δn is consistent and furthermore,
lim
n→∞ β(δ(Xn), P) = 0 ∀P ∈ P0. (7.25)
7.3 Likelihood Ratio Tests, Wald Tests, and Score Tests
We see that the Neyman-Pearson Lemma leads directly to use of the ratio of
the likelihoods in constructing tests. Now we want to generalize this approach
and to study the properties of tests based on that ratio.
There are two types of tests that arise from likelihood ratio tests. These
are called Wald tests and score tests. Score tests are also called Rao test or
Lagrange multiplier tests.
The Wald tests and score tests are asymptotically equivalent. They are
consistent under the Le Cam regularity conditions, and they are Chernoffconsistent
if α is chosen so that as n → ∞, α → 0 and χ2 r,αn ∈ o(n).
7.3.1 Likelihood Ratio Tests
Although as we have emphasized, the likelihood is a function of the distribution
rather than of the random variable, we want to study its properties
under the distribution of the random variable. Using the idea of the ratio as
in the test (7.12) of H0 : θ ∈ Θ0, but inverting that ratio and including both
hypotheses in the denominator, we define the likelihood ratio as
λ(X) = supθ∈Θ0 L(θ; X)
. (7.26)
supθ∈Θ L(θ; X)
The test, similarly to (7.12), rejects H0 if λ(X) ≤ cα, where cα is some value
in [0, 1]. Tests such as this are called likelihood ratio tests. (We should note
that there are other definitions of a likelihood ratio; in particular, in TSH3
its denominator is the sup over the alternative hypothesis. If the alternative
hypothesis does not specify Θ − Θ0, such a definition requires specification
of both H0, and H1; whereas (7.26) requires specification only of H0. Also,
the direction of the inequality depends on the ratio; it may be inverted —
compare the ratios in (7.12) and (7.26).)
The likelihood ratio may not exist, but if it is well defined, clearly it is
in the interval [0, 1], and values close to 1 provide evidence that the null
hypothesis is true, and values close to 0 provide evidence that it is false.
If there is no cα such that
Pr(λ(X) ≤ cα|H0) = α,
Theory of Statistics c○2000–2013 James E. Gentle