[Statlist] Séminaires de Statistique - Université de Neuchâtel

[KONDYLIS Atanassios]

Séminaires de Statistique 

Mardi 12-04-2005 - 11h 00 
Groupe de Statistique, Espace de l'Europe 4, Neuchâtel

Jana Jureckova
Department of Statistics, Charles University, Prague
e-mail: jurecko at karlin.mff.cuni.cz


Testing the Tail Index in Autoregressive Models
The talk is based on a joint work with Hira L. Koul (Michigan State University) and Jan Picek (Technical University in Liberec)


The study of the extreme  events such as the extreme intensity of the wind, the high flood
 levels of the rivers or extreme values of environmental  indicators, or maximal or minimal performance of a portfolio  naturally lead one to study  the tails of 
the underlying distribution rather than  its central part. Classical goodness-of-fit tests for a distribution are  usually  concerned with the central part, hence 
they cannot provide  a sufficient information on the shape of its tails. Our primary  goal is to decide whether the underlying distribution function is light- or  
heavy- tailed. The problem is semiparametric in nature,  involving an unknown slowly varying function, besides the  real-valued parameters of interest. 
A decision  in favour of a heavy tail distribution would then suggest to study the shape of the tail more closely.

 Testing the hypothesis on the tail index of a heavy tailed  distribution is an alternative inference to the classical point  estimation, surprisingly not yet much 
elaborated in the  literature. Jureckova and Picek (2001) constructed the nonparametric  tests  for the sequence of i.i.d observations. We construct a class 
of tests on the tail  index of the innovation distribution in a stationary linear autoregressive  model. The tests are nonparametric and are based on the series 
of residuals with respect to an appropriate estimator of the AR parameters; more precisely, they are based on the empirical process of maximal residuals 
of non- overlapping segments of such series. The simulation study illustrates a very good level performance of the tests. Such tests would find many 
applications in the  environmental, financial and other time series. Similar technique can be used also for time series of other  types.


References:
Jureckova, J. and J. Picek (2001). A class of tests on the tail index, Extremes, 4:2, 165--183.
Jureckova, J. (2003). Statistical tests on tail index of a probability distribution (with discussion), METRON\/  LXI/2,151--190.
Resnick, S.I. and Feigin, P.D. (1994). Limit distributions for linear programming time series estimators. J. Stoch. Process.& Appl. 51, 135-165.