Asymptotic Methods in Statistical Inference - Statistics Centre

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• Example 2. In the same situation as the previous
example, test = 0 . If the are non-normal
but are instead ∼ for in the class F of
d.f.s with mean 0 and finite variance then the
t-statistic (ˆ = ) still tends in law to Φ. Thus
( )= and the t-test is robust in its level, in
F. Simulation studies indicate that the approach
of ( )to is quite fast if is symmetric, but
can be quite slow if is skewed. Note that this
example does not contradict the theory above,
since here ˆ is consistent not only for ( 0 )=
Φ but for ( 0 ) = , when is the true
distribution.
• Example 3. The result of the previous example
(t-test) is that ( ) → for each fixed ∈
F. A stronger (and more appealing) form of robustness
requires uniformity (in )ofthisconvergence.
This fails drastically; if F is the class of
all distributions with mean 0 ,wehavethatfor
each ,
inf ( )=0and sup ( )=1