Hostname: page-component-77f85d65b8-grvzd Total loading time: 0 Render date: 2026-04-19T23:26:37.653Z Has data issue: false hasContentIssue false
  • English
  • Français

Goodness-of-fit test for long range dependent processes

Published online by Cambridge University Press:  15 November 2002

Gilles Fay  and
Anne Philippe
Gilles Fay
Affiliation:
Laboratoire de Mathématiques Appliquées, FRE 2222 du CNRS, UFR de Mathématiques, bâtiment M2, Université des Sciences et Technologies de Lille, 59655 Villeneuve-d'Ascq Cedex, France; anne.philippe@univ-lille1.fr.
Anne Philippe
Affiliation:
Laboratoire de Mathématiques Appliquées, FRE 2222 du CNRS, UFR de Mathématiques, bâtiment M2, Université des Sciences et Technologies de Lille, 59655 Villeneuve-d'Ascq Cedex, France; anne.philippe@univ-lille1.fr.
Get access

Abstract

In this paper, we make use of the information measure introducedby Mokkadem (1997) for building a goodness-of-fit test forlong-range dependent processes.Our test statistic is performed in the frequency domain and writes asa non linear functional of the normalized periodogram. We establishthe asymptotic distribution of our statistic under the nullhypothesis. Under specific alternative hypotheses, we prove that the powerconverges to one. The performance of our test procedure isillustrated from different simulated series. In particular,we compare its size and its power with test of Chenand Deo.

Keywords

Goodness-of-fit test for spectral densityperiodogramlong range dependence.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Anderson, T., Goodness of fit tests for spectral distributions. Ann. Statist. 21 (1993) 830-847. CrossRef
J.-M. Bardet, G. Lang, G. Oppenheim, A. Philippe and M. Taqqu, Generators of long-range dependent processes: A survey. Birkhäuser (2002).
M. Bartlett, An introduction to stochastic processes. Cambridge University Press (1955).
Box, G. and Pierce, D.A., Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. J. Am. Stat. Assoc. 65 (1970) 1509-1526. CrossRef
P. Brockwell and R. Davis, Time Series: Theory and Methods. Springer-Verlag, Springer Ser. in Statistics (1991).
W. Chen and R. Deo, A generalized portmanteau goodness-of-fit test for time series models. Preprint (2000).
G. Fay, Théorèmes limites pour les fonctionnelles du périodogramme, Ph.D. Thesis. École Nationale Supérieure des Télécommunications (2000).
G. Fay, E. Moulines and P. Soulier, Non linear functionals of the periodogram (submitted).
Fay, G. and Soulier, P., The periodogram of an i.i.d. sequence. Stochastic Process. Appl. 92 (2001) 315-343. CrossRef
Fox, R. and Taqqu, M., Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 (1986) 517-532. CrossRef
Giraitis, L. and Surgailis, D., 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 Fields 86 (1990) 87-104. CrossRef
U. Grenander and M. Rosenblatt, Statistical analysis of stationary time series. Wiley, New York (1957).
Hosoya, Y., A limit theory for long-range dependence and statistical inference on related models. Ann. Statist. 25 (1997) 105-137.
Hurvich, C., Moulines, E. and Soulier, P., The FEXP estimator for potentially non-stationary linear time series. Stochastic Process. Appl. 97 (2002) 307-340. CrossRef
Hurvich, C.W. and Chen, W., An efficient taper for potentially overdifferenced long-memory time series. J. Time Ser. Anal. 21 (2000) 155-180. CrossRef
Janas, D. and von Sachs, R., Consistency for non-linear functions of the periodogram of tapered data. J. Time Ser. Anal. 16 (1995) 585-606. CrossRef
Klueppelberg, C. and Mikosch, T., The integrated periodogram for stable processes. Ann. Statist. 24 (1996) 1855-1879.
Kokoszka, P. and Mikosch, T., The integrated periodogram for long-memory processes with finite or infinite variance. Stochastic Process. Appl. 66 (1997) 55-78. CrossRef
Künsch, H., Discrimination between monotonic trends and long-range dependence. J. Appl. Probab. 23 (1986) 1025-1030. CrossRef
Mikosch, T. and Norvaisa, R., Uniform convergence of the empirical spectral distribution function. Stochastic Process. Appl. 70 (1997) 85-114. CrossRef
Mokkadem, A., Une mesure d'information et son application à des tests pour les processus arma. C. R. Acad. Sci. Paris 319 (1994) 197-200.
Mokkadem, A., 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. CrossRef
Taniguchi, M., On estimation of the integrals of certain functions of spectral density. J. Appl. Probab. 17 (1980) 73-83. CrossRef
Velasco, C., Non-stationary log-periodogram regression. J. Econom. 91 (1999) 325-371. CrossRef
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.