A special integer-valued bilinear time series model with applications

Abstract

The present work proposes a special integer-valued bilinear time series model based on the thinning operators. Basic probabilistic and statistical properties of this class of models are discussed. Moreover, parameter estimation methods in the time and frequency domains and forecasting are addressed. Finally, the performances of the estimation methods are illustrated through a simulation study and an empirical application to two data sets.

Keywords

• integer-valued bilinear process
• sieve bootstrap prediction
• spectral density function
• thinning operator
• Whittle criterion

References

1. [1] B. Basrak, R.A. Davis and T. Mikosch, The sample ACF of a simple bilinear process, Stoch. Process. Their Appl. 83 (9), 1-14, 1999.
2. [2] M. Bentarzi and W. Bentarzi, Periodic integer-valued bilinear time series model, Comm. Statist. Theory Methods 46 (3), 1184-1201, 2017.
3. [3] P.J. Brockwell and R.A. Davis, Time Series: Theory and Methods, 2nd ed., Springer, 1991.
4. [4] R.A. Davis and S.I. Resnick, Limit theory for bilinear processes with heavy-tailed noise, Ann. Appl. Probab. 6 (4), 1191-1210, 1996.
5. [5] P. Doukhan, A. Latour and D. Oraichi, Simple integer-valued bilinear time series model, Adv. in Appl. Probab. 38 (2), 559-578, 2006.
6. [6] F.C. Drost, R. van den Akker and B.J.M. Werker, Note on integer-valued bilinear time series, Statist. Probab. Lett. 38 (8), 559-578, 2008.
7. [7] C.W.J. Granger and A.P. Andersen, An Introduction to Bilinear Time Series Models, Vandenhoeck and Ruprecht, Gottingen, 1978.
8. [8] M. Mohammadpour, H.S. Bakouch and S. Ramzani, An integer-valued bilinear time series model via two random operators, Math. Comput. Model. Dyn. Syst. 25 (4), 429-446, 2019.

9. [9] L. Pascual, J. Romo and E. Ruiz, Bootstrap predictive inference for ARIMA processes, J. Time Series Anal. 25 (4), 449-465, 2004.
10. [10] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1976.
11. [11] F.W. Steutel and K. van Harn, Discrete analogues of self-decomposability and stability, Ann. Probab. 7 (5), 893-899, 1979.
12. [12] T. Subba Rao, On the theory of bilinear time series models, J. R. Stat. Soc. Ser. B. Stat. Methodol. 43 (2), 244-255, 1981.
13. [13] K.F. Turkman and M.A.A. Turkman, Extremes of bilinear time series models, J. Time Series Anal. 18 (3), 305-319, 1997.
14. [14] P. Whittle, Estimation and information in stationary time series, Ark. Mat. 2 (5), 423-434, 1953.
15. [15] P. Whittle, A Study in the Analysis of Stationary Time Series, Almquist and Wiksell, Stockholm, 1954.
16. [16] Z. Zhang and H. Tong, Some distributional properties of a first-order nonnegative bilinear time series model, J. Appl. Probab. 38 (3), 659-671, 2001.