Goodness-of-fit test for long range dependent processes

Gilles Fay; Anne Philippe

ESAIM: Probability and Statistics (2010)

  • Volume: 6, page 239-258
  • ISSN: 1292-8100

Abstract

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In this paper, we make use of the information measure introduced by Mokkadem (1997) for building a goodness-of-fit test for long-range dependent processes. Our test statistic is performed in the frequency domain and writes as a non linear functional of the normalized periodogram. We establish the asymptotic distribution of our statistic under the null hypothesis. Under specific alternative hypotheses, we prove that the power converges to one. The performance of our test procedure is illustrated from different simulated series. In particular, we compare its size and its power with test of Chen and Deo.

How to cite

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Fay, Gilles, and Philippe, Anne. "Goodness-of-fit test for long range dependent processes." ESAIM: Probability and Statistics 6 (2010): 239-258. <http://eudml.org/doc/104291>.

@article{Fay2010,
abstract = { In this paper, we make use of the information measure introduced by Mokkadem (1997) for building a goodness-of-fit test for long-range dependent processes. Our test statistic is performed in the frequency domain and writes as a non linear functional of the normalized periodogram. We establish the asymptotic distribution of our statistic under the null hypothesis. Under specific alternative hypotheses, we prove that the power converges to one. The performance of our test procedure is illustrated from different simulated series. In particular, we compare its size and its power with test of Chen and Deo. },
author = {Fay, Gilles, Philippe, Anne},
journal = {ESAIM: Probability and Statistics},
keywords = {Goodness-of-fit test for spectral density; periodogram; long range dependence.},
language = {eng},
month = {3},
pages = {239-258},
publisher = {EDP Sciences},
title = {Goodness-of-fit test for long range dependent processes},
url = {http://eudml.org/doc/104291},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Fay, Gilles
AU - Philippe, Anne
TI - Goodness-of-fit test for long range dependent processes
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 239
EP - 258
AB - In this paper, we make use of the information measure introduced by Mokkadem (1997) for building a goodness-of-fit test for long-range dependent processes. Our test statistic is performed in the frequency domain and writes as a non linear functional of the normalized periodogram. We establish the asymptotic distribution of our statistic under the null hypothesis. Under specific alternative hypotheses, we prove that the power converges to one. The performance of our test procedure is illustrated from different simulated series. In particular, we compare its size and its power with test of Chen and Deo.
LA - eng
KW - Goodness-of-fit test for spectral density; periodogram; long range dependence.
UR - http://eudml.org/doc/104291
ER -

References

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  1. T. Anderson, Goodness of fit tests for spectral distributions. Ann. Statist.21 (1993) 830-847.  Zbl0779.62083
  2. J.-M. Bardet, G. Lang, G. Oppenheim, A. Philippe and M. Taqqu, Generators of long-range dependent processes: A survey. Birkhäuser (2002).  Zbl1031.65010
  3. M. Bartlett, An introduction to stochastic processes. Cambridge University Press (1955).  Zbl0068.11801
  4. G. Box and D.A. Pierce, Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. J. Am. Stat. Assoc.65 (1970) 1509-1526.  Zbl0224.62041
  5. P. Brockwell and R. Davis, Time Series: Theory and Methods. Springer-Verlag, Springer Ser. in Statistics (1991).  Zbl0709.62080
  6. W. Chen and R. Deo, A generalized portmanteau goodness-of-fit test for time series models. Preprint (2000).  Zbl1072.62088
  7. G. Fay, Théorèmes limites pour les fonctionnelles du périodogramme, Ph.D. Thesis. École Nationale Supérieure des Télécommunications (2000).  
  8. G. Fay, E. Moulines and P. Soulier, Non linear functionals of the periodogram (submitted).  Zbl1063.62015
  9. G. Fay and P. Soulier, The periodogram of an i.i.d. sequence. Stochastic Process. Appl.92 (2001) 315-343.  Zbl1046.62097
  10. R. Fox and M. Taqqu, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist.14 (1986) 517-532.  Zbl0606.62096
  11. L. Giraitis and D. Surgailis, A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittles's estimate. Probab. Theory Related Fields86 (1990) 87-104.  Zbl0717.62015
  12. U. Grenander and M. Rosenblatt, Statistical analysis of stationary time series. Wiley, New York (1957).  Zbl0080.12904
  13. Y. Hosoya, A limit theory for long-range dependence and statistical inference on related models. Ann. Statist.25 (1997) 105-137.  Zbl0873.62096
  14. C. Hurvich, E. Moulines and P. Soulier, The FEXP estimator for potentially non-stationary linear time series. Stochastic Process. Appl.97 (2002) 307-340.  Zbl1057.62074
  15. C.W. Hurvich and W. Chen, An efficient taper for potentially overdifferenced long-memory time series. J. Time Ser. Anal.21 (2000) 155-180.  Zbl0958.62085
  16. D. Janas and R. von Sachs, Consistency for non-linear functions of the periodogram of tapered data. J. Time Ser. Anal.16 (1995) 585-606.  Zbl0852.62088
  17. C. Klueppelberg and T. Mikosch, The integrated periodogram for stable processes. Ann. Statist.24 (1996) 1855-1879.  Zbl0898.62116
  18. P. Kokoszka and T. Mikosch, The integrated periodogram for long-memory processes with finite or infinite variance. Stochastic Process. Appl.66 (1997) 55-78.  Zbl0885.62108
  19. H. Künsch, Discrimination between monotonic trends and long-range dependence. J. Appl. Probab.23 (1986) 1025-1030.  Zbl0623.62085
  20. T. Mikosch and R. Norvaisa, Uniform convergence of the empirical spectral distribution function. Stochastic Process. Appl.70 (1997) 85-114.  Zbl0913.60032
  21. A. Mokkadem, Une mesure d'information et son application à des tests pour les processus arma. C. R. Acad. Sci. Paris319 (1994) 197-200.  Zbl0801.62082
  22. A. Mokkadem, A measure of information and its applications to test for randomness against ARMA alternatives and to goodness-of-fit test. Stochastic Process. Appl.72 (1997) 145-159.  Zbl0936.62101
  23. M. Taniguchi, On estimation of the integrals of certain functions of spectral density. J. Appl. Probab.17 (1980) 73-83.  Zbl0428.62055
  24. C. Velasco, Non-stationary log-periodogram regression. J. Econom.91 (1999) 325-371.  Zbl1041.62533
  25. Y. Yajima, Asymptotic properties of estimates in incorrect ARMA models for long-memory time series, in New directions in time series analysis. Part II. Proc. Workshop, Minneapolis/MN (USA) 1990. Springer, New York, IMA Vol. Math. Appl. 46 (1993) 375-382.  Zbl0767.62075

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