Goodness-of-fit test for long range dependent processes
ESAIM: Probability and Statistics, Tome 6 (2002), pp. 239-258.

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.

DOI : 10.1051/ps:2002013
Classification : 60F05, 62F03
Mots clés : goodness-of-fit test for spectral density, periodogram, long range dependence 
@article{PS_2002__6__239_0,
     author = {Fay, Gilles and Philippe, Anne},
     title = {Goodness-of-fit test for long range dependent processes},
     journal = {ESAIM: Probability and Statistics},
     pages = {239--258},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2002},
     doi = {10.1051/ps:2002013},
     mrnumber = {1943149},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2002013/}
}
TY  - JOUR
AU  - Fay, Gilles
AU  - Philippe, Anne
TI  - Goodness-of-fit test for long range dependent processes
JO  - ESAIM: Probability and Statistics
PY  - 2002
SP  - 239
EP  - 258
VL  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2002013/
DO  - 10.1051/ps:2002013
LA  - en
ID  - PS_2002__6__239_0
ER  - 
%0 Journal Article
%A Fay, Gilles
%A Philippe, Anne
%T Goodness-of-fit test for long range dependent processes
%J ESAIM: Probability and Statistics
%D 2002
%P 239-258
%V 6
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps:2002013/
%R 10.1051/ps:2002013
%G en
%F PS_2002__6__239_0
Fay, Gilles; Philippe, Anne. Goodness-of-fit test for long range dependent processes. ESAIM: Probability and Statistics, Tome 6 (2002), pp. 239-258. doi : 10.1051/ps:2002013. http://archive.numdam.org/articles/10.1051/ps:2002013/

[1] T. Anderson, Goodness of fit tests for spectral distributions. Ann. Statist. 21 (1993) 830-847. | MR | Zbl

[2] J.-M. Bardet, G. Lang, G. Oppenheim, A. Philippe and M. Taqqu, Generators of long-range dependent processes: A survey. Birkhäuser (2002). | MR | Zbl

[3] M. Bartlett, An introduction to stochastic processes. Cambridge University Press (1955). | MR | Zbl

[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. | MR | Zbl

[5] P. Brockwell and R. Davis, Time Series: Theory and Methods. Springer-Verlag, Springer Ser. in Statistics (1991). | MR | Zbl

[6] W. Chen and R. Deo, A generalized portmanteau goodness-of-fit test for time series models. Preprint (2000). | MR | Zbl

[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). | Zbl

[9] G. Fay and P. Soulier, The periodogram of an i.i.d. sequence. Stochastic Process. Appl. 92 (2001) 315-343. | MR | Zbl

[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. | MR | Zbl

[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 Fields 86 (1990) 87-104. | Zbl

[12] U. Grenander and M. Rosenblatt, Statistical analysis of stationary time series. Wiley, New York (1957). | MR | Zbl

[13] Y. Hosoya, A limit theory for long-range dependence and statistical inference on related models. Ann. Statist. 25 (1997) 105-137. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[17] C. Klueppelberg and T. Mikosch, The integrated periodogram for stable processes. Ann. Statist. 24 (1996) 1855-1879. | MR | Zbl

[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. | MR | Zbl

[19] H. Künsch, Discrimination between monotonic trends and long-range dependence. J. Appl. Probab. 23 (1986) 1025-1030. | MR | Zbl

[20] T. Mikosch and R. Norvaisa, Uniform convergence of the empirical spectral distribution function. Stochastic Process. Appl. 70 (1997) 85-114. | MR | Zbl

[21] A. Mokkadem, Une mesure d'information et son application à des tests pour les processus arma. C. R. Acad. Sci. Paris 319 (1994) 197-200. | Zbl

[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. | MR | Zbl

[23] M. Taniguchi, On estimation of the integrals of certain functions of spectral density. J. Appl. Probab. 17 (1980) 73-83. | MR | Zbl

[24] C. Velasco, Non-stationary log-periodogram regression. J. Econom. 91 (1999) 325-371. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :