Exact Converegence Rate and Leading Term in Central Limit Theorem for
Student's t Statistic
Peter Hall and Qiying Wang
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 Studentised 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-Studentised
sum when the
latter is normalised for scale using a truncated form of variance, but
when the corresponding truncated centring constant is omitted. Examples of
characterisations 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.
AMS Classification:Primary 60F15,
Secondary 62E20 Date: 9 January 2003
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