MatStat seminarium tisdagen, den 5/12 kl 10.15

Title: Applying Extreme Value Theory to Wavelet Transforms

Marc Raimondo, University of Sydney

ABSTRACT

 Newly available wavelet bases on multi-resolution analysis have
 exiting implications for detection of change-points. By checking the
 absolute value of wavelet coefficients one can detect discontinuities
 in a smooth curve even in the presence of additive noise. In this
 talk, we combine wavelet methods and extreme value theory to test the
 presence of an arbitrary number of discontinuities in a unknown
 function observed with noise. We present a test statistic which is
 based on a ``translation-invariant''wavelet transformation (IWT).
 Particular feature of our approach is to use a Peaks Over Threshold
 (POT) model to derive the limiting distribution of the test
 statistic. Assuming that the noise distribution belongs to the domain
 of attraction of an Extreme Value Distribution; we show that our
 decision rule is asymptotically optimal among all tests which satisfy
 a size restriction. Practical implementation of our method includes
 consistent estimators of the shape and scale parameters in the tail
 of noise distribution as well as of the number and the location of
 the discontinuities in the original signal. We compare our test with
 competing approaches on simulated examples and conclude that our
 procedure outperforms existing methods.  We illustrate our method on
 the Dow-Jones data.