Bootstrapping periodically autoregressive models

Ciołek, Gabriela; Potorski, Paweł

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 394-411.

The main objective of this paper is to establish the residual and the wild bootstrap procedures for periodically autoregressive models. We use the least squares estimators of model’s parameters and generate their bootstrap equivalents. We prove that the bootstrap procedures for causal periodic autoregressive time series with finite fourth moments are weakly consistent. Finally, we confirm our theoretical considerations by simulations.

Classification : 62M10, 62F12, 62F40
Mots clés : Bootstrap, least squares estimation, periodically autoregressive models, time series

R. Arora, W. Sethares and J. Bucklew, Latent periodicities in genome sequences. IEEE J. Selected Topics Signal Proc. 2 (2008) 332–342. | DOI

I.V. Basawa and R.B. Lund, Recursive prediction and likelihood evaluation for periodic ARMA models. J. Time Ser. Ana. 21 (2000) 75–93. | DOI | MR | Zbl

I.V. Basawa and R.B. Lund, Large sample properties of parameter for periodic ARMA models. J. Time Ser. Anal. 21 (2001) 75–93. | MR

A. Bibi and I. Lescheb, A note on integrated periodic GARCH processes. Statist. Probab. Lett. 87 (2014) 121–124. | DOI | MR | Zbl

E. Broszkiewicz-Suwaj, A. Makagon, R. Weron and A. Wyłomańska, On detecting and modeling periodic correlation in financial data. Proceedings of the XVIII Max Born Symposium. Phys. A: Statist. Mech. Appl. 336 (2004) 196–205. | DOI | MR

E. Carlstein, The use of subseries values for estimating the variance of a general statistics from a stationary sequence. Ann. Statist. 14 (1986) 1171–1179. | DOI | MR | Zbl

V. Chan, S.N. Lahiri and W.Q. Meeker, Block bootstrap estimation of the distribution of cumulative outdoor degradation. Technometrics 46 (2004) 215–224. | DOI | MR

J. Dowell, S. Weiss, D. Hill and D. Infield, A cyclo-stationary complex multichannel Wiener filter for the prediction of wind speed and direction. Proc. 21st Europ. Signal Proc. Confer. (EUSIPCO 2013).

A.E. Dudek, J. Leśkow, E. Paparoditis and D. Politis, A generalized block bootstrap for seasonal time series. J. Time Ser. Anal. 35 (2014) 89–114. | DOI | MR | Zbl

W. Gardner and L.E. Franks, Characterization of cyclostationary random signal processes. IEEE Trans. Inform. Theory 21 (1975) 4–14. | DOI | Zbl

C. Gaucherel, Analysis of ENSO interannual oscillations using non-stationary quasi-periodic statistics. A study of ENSO memory. Int. J. Climatology 30 (2010) 926–934. | DOI

F. Ghaderi, K. Nazarpour, J.G. Mcwhirter and S. Sanei, Removal of ballistocardiogram artifacts using the cyclostationary source extraction method. IEEE Trans. Biomed. Eng. 57 (2010) 2667–2676. | DOI

E.G. Gladyšhev, Periodically correlated random sequences. Soviet Math. 2 (1961) 385–388. | MR | Zbl

P. Hall, Resampling a coverage pattern. Stoch. Process. Appl. 20 (1985) 231–246. | DOI | MR | Zbl

P. Hall, The Bootstrap and Edgeworth Expansion. New York: Springer Verlag (1992). | MR | Zbl

J.D. Hamilton, Time Series Anal. New Jersey, Princeton University Press (1994). | MR | Zbl

B. Iqelan, Periodically Correlated Time Series: Models and Examples. Lambert Academic Publishing (2011).

R.H. Jones and W.M. Brelsford, Time series with periodic structure. Biometrika 54 (1967) 403–408. | DOI | MR | Zbl

K.-Y. Kim, B. Hamlington and H. Na, Theoretical foundation of cyclostationary EOF analysis for geophysical and climatic variables. Concepts and examples. Earth-Sci. Rev. 150 (2015) 201–218. | DOI

J. Kreiss and S. Lahiri, Bootstrap Methods for Time Series. Time Ser. Anal.: Methods Appl. 30 (2012) 3–23.

H. Künsch, The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 (1989) 1217–1241. | DOI | MR | Zbl

S. Lahiri, Resampling methods for Dependent Data. Springer Verlag (2003). | MR | Zbl

R. Liu, Bootstrap Procedures under some Non-I.I.D. Models. Ann. Statist. 164 (1988) 1696–1708. | MR | Zbl

R.Y. Liu and K. Singh, Moving blocks jackknife and bootstrap capture weak dependence. Exploring the Limits of Bootstrap, edited by R. LePage and L. Billard. Wiley New York (1992) 225–248. | MR | Zbl

S. Maiz, El. Badaoui, F. Bonnardot and C. Serviere, New second order cyclostationary analysis and application to the detection and characterization of a runner’s fatigue. Signal Processing 102 (2014) 188–200. | DOI

A.S. Monin, Stationary and periodic time series in the general circulation of the atmosphere, in: Proc. Symp. Time Ser. Anal., edited by M. Rosenblatt. John Wiley and Sons, New York (1963) 144–151. | MR

M. Pagano, On periodic and multiple autoregressions. Ann. Statist. 6 (1978) 1310–1317. | DOI | MR | Zbl

E. Parzen and M. Pagano, An approach to modeling seasonally stationary time series. J. Econom. 9 (1979) 137–153. | DOI | Zbl

D.N. Politis, Resampling time series with seasonal components, in Frontiers in Data Mining and Bioinformatics: Proceedings of the 33rd Symposium on the Interface of Computing Science and Statistics, Orange County, California 13-17 (2001) 619–621.

Q. Shao and P.P. Ni, Least-squares estimation and ANOVA for periodic autoregressive time series. Statist. Probab. Lett. 69 (2004) 287–297. | DOI | MR | Zbl

K. Shimizu, Boostrapping stationary ARMA-GARCH Models. Vieweg$+$Teubner Research (2009).

G.B. Thomas and D.F. Fiering, Mathematical synthesis of streamflow sequences for the analysis of river basins by simulation, in: Design of Water Resources Syst., edited by A. Maas. Harvard University Press, Cambridge (1962).

E. Ursu and P. Duchesne, On modelling and diagnostic checking of vector periodic autoregressive time series models. J. Time Seri. Anal. 30 (2009) 70–96. | DOI | MR | Zbl

A. Vecchia, Maximum Likelihood Estimation for Periodic Autoregressive Moving Average Models. Technometrics 27 (1985) 375–384. | DOI