Asymptotic Methods in Statistical Inference - Statistics Centre

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• The three tests are asymptotically equivalent. The
Scores test has the advantage of not requiring
the computation of ˆ . The Scores and LR tests
are invariant under reparameterizations; i.e. they
have the same value if one tests : = 0 for a
1 − 1(differentiable) function = (). Wald’s
test has this property (in finite samples) only for
linear transformations (nonlinear transformations
as well, asymptotically). You should verify these
statements.
• The behaviour under can be quite different.
See Example 7.7.3 for instance, where for the
Scores test of a Cauchy median , thepower→ 0
as | − 0 | →∞for fixed . There are analogous
examples with respect to Wald’s test. The
LR test is most often preferable in terms of power.