Abstract
The leading term in the normal approximation to the distribution of Student's
t statistic is derived in a general setting, with the sole assumption being
that the sampled distribution is in the domain of attraction of a normal law.
The form of the leading term is shown to have its origin in the way in which
extreme data influence properties of the Studentized sum. The leading-term
approximation is used to give the exact rate of convergence in the central
limit theorem up to order n^-1/2, where n denotes sample size. It is proved
that the exact rate uniformly on the whole real line is identical to the exact
rate on sets of just three points. Moreover, the exact rate is identical to
that for the non-Studentized sum when the latter is normalized for scale using
a truncated form of variance, but when the corresponding truncated centering
constant is omitted. Examples of characterizations of convergence rates are
also given. It is shown that, in some instances, their validity uniformly on
the whole real line is equivalent to their validity on just two symmetric
points.
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