ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 2, pp. 119–137.
Pleiades Publishing, Inc., 2008.
Original Russian Text
M.S. Ermakov, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 2, pp. 54–74.
METHODS OF SIGNAL PROCESSING
Nonparametric Hypothesis Testing with Small
Type I or Type II Error Probabilities
M. S. Ermakov
Institute of Problems of Mechanical Engineering, RAS, St. Petersburg
Received June 13, 2007; in ﬁnal form, March 13, 2008
Abstract—For the problem of signal detection in Gaussian white noise, we obtain lower bounds
for the asymptotics of moderate deviation probabilities of type I and type II errors. These
asymptotics are attained on tests of the χ
type. Using these lower bounds, we ﬁnd lower
bounds for nonparametric conﬁdence estimation in the moderate deviation zone.
In hypothesis testing, it is usually assumed that type I error probabilities are small. Therefore,
along with the generally used comparison of test quality under ﬁxed type I error probabilities, test
quality is also studied for type I error probabilities tending to zero. This setup corresponds to
considering the problem in terms of large and moderate deviation probabilities.
The most commonly used measures of eﬃciency in large and moderate deviation zones are,
respectively, the Bahadur  and Kallenberg  eﬃciencies. These eﬃciency measures compare
the quality of tests based on their type I error probabilities assuming that their type II error
probabilities are separated from zero and one. In [3–5] it was proved that in parametric hypothesis
testing problems, the local Bahadur, local Chernoﬀ , and local Hodges–Lehmann  eﬃciencies
in fact coincide with the Pitman eﬃciency. In [8, 9] it was shown that in the moderate deviation
zone, the Pitman eﬃciency remains a natural eﬃciency measure for almost arbitrary variations
of type I and type II error probabilities. The results were based on the analysis of new types
of eﬃciency for logarithmic and strong asymptotics of moderate deviation probabilities of type I
and type II errors (see [8, 9]). These eﬃciencies were called the MD (moderate deviation) and
SMD (strong moderate deviation) eﬃciencies, respectively. In , for a similar problem setting,
analogous versions of eﬃciencies were called conservation laws.
The goal of the present paper is to study the MD and SMD eﬃciencies in the problems of
testing nonparametric hypotheses. Here we dwell on the problem of nonparametric signal detec-
tion in Gaussian white noise, since the theory of asymptotic equivalence of statistical experiments
(see [11, 12]) makes it possible to reduce a large number of other problems of nonparametric hy-
pothesis testing to this problem. The technique is based on the Pinsker approach to proving lower
bounds in nonparametric estimation (see [13, 14]). In , it was extended to problems of testing
nonparametric hypotheses. It should be noted that in recent years the theory of nonparametric
signal detection in Gaussian white noise was intensively developed. The most complete overview
of results on this subject can be found in .
While studying the behavior of classical parametric and nonparametric tests from the viewpoint
of large and moderate deviation probabilities has become a traditional research subject (see ),
testing of nonparametric hypotheses taking into account a priori information on smoothness of