Statistica Sinica 19 (2009), 343-354
RELATIVE ERRORS IN
CENTRAL LIMIT THEOREMS
FOR
STUDENT'S 
STATISTIC, WITH APPLICATIONS
Qiying Wang and Peter Hall
University of Sydney and University of
Melbourne
Abstract: Student's statistic is frequently used
in practice to test hypotheses about means.
Today, in fields such as genomics, tens of thousands of -tests are implemented
simultaneously, one for each component of a long data vector.
The distributions from which the statistics are computed are almost
invariably non-normal and skew, and the sample sizes are relatively small,
typically about one thousand times smaller than the number of tests.
Therefore, theoretical investigations of the accuracy of the tests would be
based on large-deviation expansions. Recent research has shown that in this setting,
unlike classical contexts, weak dependence among vector components is
often not a problem; independence can safely be assumed when the significance
level is very small, provided dependence among the test statistics is short range.
However, conventional large-deviation results provide information only about
the accuracy of normal and Student's approximations under the null hypothesis.
Power properties, especially against sparse local alternatives,
require more general expansions where the data no longer have zero mean,
and in fact where the mean can depend on both sample size and the number
of tests. In this paper we derive this type of expansion, and show how it
can be used to draw statistical conclusions about the effectiveness of
many simultaneous -tests. Similar arguments can be used to derive properties
of classifiers based on high-dimensional data.
Key words and phrases: Genomics,
classification, family-wise error rate, large-deviation expansion,
signal detection, simultaneous hypothesis testing, sparsity.