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BAHADUR EFFICIENCY AND ROBUSTNESS OF

STUDENTIZED SCORE TESTS

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XUMING HE^{ 1} AND QI-MAN
SHAO^{ 2}

^{1} *Department of Statistics, University of Illinois,
Champaign,
IL 61820, U.S.A.*

^{2} *Department of Mathematics, National University of
Singapore,
Singapore 0511*
(Received July 27, 1994; revised July 14, 1995)

**Abstract.**
We derive the exact Bahadur slopes of
studentized score tests for a simple null hypothesis in a
one-parameter family of distributions. The Student's *t*-test is
included as a special case for which a recent result of Rukhin
(1993, *Sankhyã Ser. A*, **55**, 159-163) was
improved upon. It is shown that locally optimal Bahadur efficiency
for one-sample location models with a known or estimated scale
parameter is attained within the class of studentized score tests.
The studentized test has an asymptotic null distribution free of the
scale parameter, and the optimality of likelihood scores does not
depend on the existence of a moment generating function. We also
consider the influence function and breakdown point of such tests as
part of our robustness investigation. The influence of any
studentized score test is bounded from above, indicating certain
degree of robustness of validity, but a bounded score function is
needed to cap the influence from below and to ensure a high power
breakdown point. We find that the standard Huber-type score tests are
not only locally minimax in Bahadur efficiency, but also very
competitive in global efficiency at a variety of location models.

*Key words and phrases*:
Bahadur slope, efficiency,
influence function, score test.

**Source**
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