 
Summary: Volume 14, No. 3 (2005), pp. Allerton Press, Inc.
M A T H E M A T I C A L M E T H O D S O F S T A T I S T I C S
LOCAL FUNCTIONAL HYPOTHESIS TESTING
F. Abramovich and R. Heller
Dept. Statistics and Operations Research, Tel Aviv University
Tel Aviv 69978, Israel
Email: felix@post.tau.ac.il, rheller@post.tau.ac.il
We consider a standard "signal+white noise" model on the unit interval and want to test
whether the signal is present on a subinterval [0, 1] of length . The composite alternative
is that the unknown signal f is separated away from zero in terms of its average power (f) =
f 2
/ on and also possesses some regularity properties. We evaluate the asymptotically
optimal (minimax) rates for testing the presence of a signal on , where both the noise level
and the interval length tend to zero. We derive corresponding rateoptimal tests for local signal
detection.
Key words: adaptive testing, Besov spaces, functional hypothesis testing, local signal detection,
minimax hypothesis testing, wavelets.
2000 Mathematics Subject Classification: primary 62G10, secondary 62G20.
1. Introduction
Consider a standard "signal + white noise" model, where the data is a sample
