Abstract
Original language  English 

Qualification  PhD 
Awarding Institution 

Supervisors/Advisors 

Thesis sponsors  
Award date  4 Dec 2018 
Publication status  Published  11 Jan 2019 
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Keywords
 Time series
 Noncausal process
 Anticipative process
 Multivariate stable distribution
 Conditional distribution
 Prediction
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Anticipative alphastable linear processes for time series analysis : conditional dynamics and estimation. / Fries, Sebastien.
2019. 239 p.Research output: PhD Thesis › PhD Thesis  Research external, graduation external › Academic
TY  THES
T1  Anticipative alphastable linear processes for time series analysis : conditional dynamics and estimation
AU  Fries, Sebastien
PY  2019/1/11
Y1  2019/1/11
N2  In the framework of linear time series analysis, we study a class of socalled anticipative strictly stationary processes potentially depending on all the terms of an independent and identically distributed alphastable errors sequence.Focusing first on autoregressive (AR) processes, it is shown that higher order conditional moments than marginal ones exist provided the characteristic polynomials admits at least one root inside the unit circle. The forms of the first and second order moments are obtained in special cases.The least squares method is shown to provide a consistent estimator of an allpass causal representation of the process, the validity of which can be tested by a portmanteautype test. A method based on extreme residuals clustering is proposed to determine the original AR representation.The anticipative stable AR(1) is studied in details in the framework of bivariate alphastable random vectors and the functional forms of its first four conditional moments are obtained under any admissible parameterisation.It is shown that during extreme events, these moments become equivalent to those of a twopoint distribution charging two polarlyopposite future paths: exponential growth or collapse.Parallel results are obtained for the continuous time counterpart of the AR(1), the anticipative stable OrnsteinUhlenbeck process.For infinite alphastable moving averages, the conditional distribution of future paths given the observed past trajectory during extreme events is derived on the basis of a new representation of stable random vectors on unit cylinders relative to seminorms.Contrary to the case of norms, such representation yield a multivariate regularly varying tails property appropriate for prediction purposes, but not all stable vectors admit such a representation.A characterisation is provided and it is shown that finite length paths of a stable moving average admit such representation provided the process is "anticipative enough".Processes resulting from the linear combination of stable moving averages are encompassed, and the conditional distribution has a natural interpretation in terms of pattern identification.
AB  In the framework of linear time series analysis, we study a class of socalled anticipative strictly stationary processes potentially depending on all the terms of an independent and identically distributed alphastable errors sequence.Focusing first on autoregressive (AR) processes, it is shown that higher order conditional moments than marginal ones exist provided the characteristic polynomials admits at least one root inside the unit circle. The forms of the first and second order moments are obtained in special cases.The least squares method is shown to provide a consistent estimator of an allpass causal representation of the process, the validity of which can be tested by a portmanteautype test. A method based on extreme residuals clustering is proposed to determine the original AR representation.The anticipative stable AR(1) is studied in details in the framework of bivariate alphastable random vectors and the functional forms of its first four conditional moments are obtained under any admissible parameterisation.It is shown that during extreme events, these moments become equivalent to those of a twopoint distribution charging two polarlyopposite future paths: exponential growth or collapse.Parallel results are obtained for the continuous time counterpart of the AR(1), the anticipative stable OrnsteinUhlenbeck process.For infinite alphastable moving averages, the conditional distribution of future paths given the observed past trajectory during extreme events is derived on the basis of a new representation of stable random vectors on unit cylinders relative to seminorms.Contrary to the case of norms, such representation yield a multivariate regularly varying tails property appropriate for prediction purposes, but not all stable vectors admit such a representation.A characterisation is provided and it is shown that finite length paths of a stable moving average admit such representation provided the process is "anticipative enough".Processes resulting from the linear combination of stable moving averages are encompassed, and the conditional distribution has a natural interpretation in terms of pattern identification.
KW  Time series
KW  Noncausal process
KW  Anticipative process
KW  Multivariate stable distribution
KW  Conditional distribution
KW  Prediction
M3  PhD Thesis  Research external, graduation external
ER 