QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations

Abdelouahab Bibi; Ahmed Ghezal

Kybernetika (2018)

  • Volume: 54, Issue: 2, page 375-399
  • ISSN: 0023-5954

Abstract

top
This paper develops an asymptotic inference theory for bilinear B L time series models with periodic coefficients P B L for short . For this purpose, we establish firstly a necessary and sufficient conditions for such models to have a unique stationary and ergodic solutions (in periodic sense). Secondly, we examine the consistency and the asymptotic normality of the quasi-maximum likelihood estimator Q M L E under very mild moment condition for the innovation errors. As a result, it is shown that whenever the model is strictly stationary, the moment of some positive order of P B L model exists and is finite, under which the strong consistency and asymptotic normality of Q M L E for P B L are proved. Moreover, we consider also the periodic A R M A P A R M A models with P B L innovations and we prove the consistency and the asymptotic normality of its Q M L E .

How to cite

top

Bibi, Abdelouahab, and Ghezal, Ahmed. "QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations." Kybernetika 54.2 (2018): 375-399. <http://eudml.org/doc/294733>.

@article{Bibi2018,
abstract = {This paper develops an asymptotic inference theory for bilinear $\left( BL\right) $ time series models with periodic coefficients $\left( PBL\text\{ for short\}\right) $. For this purpose, we establish firstly a necessary and sufficient conditions for such models to have a unique stationary and ergodic solutions (in periodic sense). Secondly, we examine the consistency and the asymptotic normality of the quasi-maximum likelihood estimator $\left( QMLE\right) $ under very mild moment condition for the innovation errors. As a result, it is shown that whenever the model is strictly stationary, the moment of some positive order of $PBL$ model exists and is finite, under which the strong consistency and asymptotic normality of $QMLE$ for $PBL$ are proved. Moreover, we consider also the periodic $ARMA$$\left( PARMA\right) $ models with $PBL$ innovations and we prove the consistency and the asymptotic normality of its $QMLE$.},
author = {Bibi, Abdelouahab, Ghezal, Ahmed},
journal = {Kybernetika},
keywords = {periodic bilinear model; periodic $ARMA$ model; strict and second-order periodic stationarity; strong consistency; asymptotic normality},
language = {eng},
number = {2},
pages = {375-399},
publisher = {Institute of Information Theory and Automation AS CR},
title = {QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations},
url = {http://eudml.org/doc/294733},
volume = {54},
year = {2018},
}

TY - JOUR
AU - Bibi, Abdelouahab
AU - Ghezal, Ahmed
TI - QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations
JO - Kybernetika
PY - 2018
PB - Institute of Information Theory and Automation AS CR
VL - 54
IS - 2
SP - 375
EP - 399
AB - This paper develops an asymptotic inference theory for bilinear $\left( BL\right) $ time series models with periodic coefficients $\left( PBL\text{ for short}\right) $. For this purpose, we establish firstly a necessary and sufficient conditions for such models to have a unique stationary and ergodic solutions (in periodic sense). Secondly, we examine the consistency and the asymptotic normality of the quasi-maximum likelihood estimator $\left( QMLE\right) $ under very mild moment condition for the innovation errors. As a result, it is shown that whenever the model is strictly stationary, the moment of some positive order of $PBL$ model exists and is finite, under which the strong consistency and asymptotic normality of $QMLE$ for $PBL$ are proved. Moreover, we consider also the periodic $ARMA$$\left( PARMA\right) $ models with $PBL$ innovations and we prove the consistency and the asymptotic normality of its $QMLE$.
LA - eng
KW - periodic bilinear model; periodic $ARMA$ model; strict and second-order periodic stationarity; strong consistency; asymptotic normality
UR - http://eudml.org/doc/294733
ER -

References

top
  1. Aknouche, A., Bibi, A., 10.1111/j.1467-9892.2008.00598.x, J. Time Ser. Anal. 30 (2008), 1, 19-46. MR2488634DOI10.1111/j.1467-9892.2008.00598.x
  2. Aknouche, A., Guerbyenne, H., 10.1016/j.spl.2008.12.012, Statist. Probab. Lett. 79 (2009), 7, 990-996. MR2509491DOI10.1016/j.spl.2008.12.012
  3. Aknouche, A., Bibi, A., 10.1111/j.1467-9892.2008.00598.x, J. Time Ser. Anal. 30 (2008), 1, 19-46. MR2488634DOI10.1111/j.1467-9892.2008.00598.x
  4. Basawa, I. V., Lund, R., 10.1111/1467-9892.00246, J. Time Series Anal. 22 (2001), 6, 651-663. MR1867391DOI10.1111/1467-9892.00246
  5. Bibi, A., 10.1081/sap-120017531, Stochastic Anal. App. 21 (2003), 1, 25-60. MR1954074DOI10.1081/sap-120017531
  6. Bibi, A., Francq, C., 10.1007/bf02530484, Ann. Inst. Statist. Math. 55 (2003), 1, 41-68. MR1965962DOI10.1007/bf02530484
  7. Bibi, A., Oyet, A. J., 10.1081/sap-120028595, Stochastic Anal. Appl. 22 (2004), 2, 355-376. MR2037377DOI10.1081/sap-120028595
  8. Bibi, A., Aknouche, A., 10.1007/s10260-008-0110-z, Statist. Methods Appl. 19 (2010), 1, 1-30. MR2591755DOI10.1007/s10260-008-0110-z
  9. Bibi, A., Lessak, R., 10.1016/j.spl.2008.07.024, Statist. Probab. Lett. 79 (2009), 1, 79-87. MR2483399DOI10.1016/j.spl.2008.07.024
  10. Bibi, A., Lescheb, I., 10.1016/j.econlet.2011.11.013, Econom. Lett. 114 (2012), 3, 353-357. MR2880617DOI10.1016/j.econlet.2011.11.013
  11. Bibi, A., Ghezal, A., 10.1007/s10260-015-0344-5, Stat. Methods Appl. 25 (2016), 3, 395-420. MR3539499DOI10.1007/s10260-015-0344-5
  12. Billingsley, P., Probability and Measure. Third edition., Wiley - Interscience 1995. MR1324786
  13. Bougerol, P., Picard, N., 10.1214/aop/1176989526, Ann. Probab. 20 (1992), 4, 1714-1730. MR1188039DOI10.1214/aop/1176989526
  14. Boyles, R. A., Gardner, W. A., 10.1109/tit.1983.1056613, IEEE, Trans. Inform. Theory 29 (1983), 105-114. MR0711279DOI10.1109/tit.1983.1056613
  15. Brandt, A., 10.2307/1427243, Adv. Appl. Probab. 18 (1986), 1, 211-220. MR0827336DOI10.2307/1427243
  16. Chatterjee, S., Das, S., 10.1081/sta-120021324, Comm. Stat. Theory Methods 32 (2003), 6, 1135-1153. MR1983236DOI10.1081/sta-120021324
  17. Florian, Z., 10.1007/978-3-319-13881-7_23, Time Series. Stochastic Models, Statistics and their Applications 122 (2015), 207-214. DOI10.1007/978-3-319-13881-7_23
  18. Francq, C., Roy, R., Saidi, A., 10.1111/j.1467-9892.2011.00728.x, J. of Time Ser. Anal. 32 (2011), 699-723. MR2846568DOI10.1111/j.1467-9892.2011.00728.x
  19. Francq, C., 10.1080/15326349908807524, Comm. Statist. Stochastic Models 15 (1999), 1, 29-52. MR1674085DOI10.1080/15326349908807524
  20. Francq, C., Zakoîan, J. M., 10.3150/bj/1093265632, Bernoulli 10 (2004), 4, 605-637. MR2076065DOI10.3150/bj/1093265632
  21. Gardner, W. A., Nopolitano, A., Paura, L., 10.1016/j.sigpro.2005.06.016, Signal Process. 86 (2006), 4, 639-697. DOI10.1016/j.sigpro.2005.06.016
  22. Gladyshev, E. G., Periodically correlated random sequences., Soviet Math. 2 (1961), 385-388. MR0126873
  23. Grahn, T., 10.1111/j.1467-9892.1995.tb00251.x, J. Time Series Analysis 16 (1995), 509-529. MR1365644DOI10.1111/j.1467-9892.1995.tb00251.x
  24. He, J., Yu, S., Cai, J., 10.1142/s0218127416502199, Int. J. Bifurcation Chaos 26 (2016), 13, 1-11. MR3590146DOI10.1142/s0218127416502199
  25. Kesten, H., Spitzer, F., 10.1007/bf00532045, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 67 (1984), 363-386. MR0761563DOI10.1007/bf00532045
  26. Kristensen, D., 10.1111/j.1467-9892.2008.00603.x, J. Time Series Anal. 30 (2009), 1, 125-144. MR2488638DOI10.1111/j.1467-9892.2008.00603.x
  27. Ling, S., Peng, L., Zho, F., Inference for special bilinear time series model., arXiv 2014. MR3300205
  28. Liu, J., 10.1080/15326349908807167, Stochastic Models 6 (1990), 649-665. MR1080415DOI10.1080/15326349908807167
  29. Ngatchou-W, J., 10.1214/07-ejs157, Elec. J. Stat. 2 (2008), 40-62. MR2386085DOI10.1214/07-ejs157
  30. Pan, J. Z., Li, G. D., Xie, Z.J., Stationary solution and parametric estimation for bilinear model driven by A R C H noises., Science in China (Series ) 45 (2002), 12, 1523-1537. MR1955672
  31. Pham, D. T., 10.1016/0304-4149(85)90216-9, Stoch. Proc. Appl. 20 (1983), 295-306. MR0808163DOI10.1016/0304-4149(85)90216-9
  32. Rao, T. Subba, Gabr, M. M., 10.1007/978-1-4684-6318-7, Lecture Notes In Statistics 24 (1984), Springer Verlag, N.Y. MR0757536DOI10.1007/978-1-4684-6318-7
  33. Tjøstheim, D., 10.1016/0304-4149(86)90099-2, Stoch. Proc. Appl. 21 (1986), 251-273. MR0833954DOI10.1016/0304-4149(86)90099-2
  34. Wittwer, G. S., 10.1080/02331888908802201, Statistics 20 (1989), 521-529. MR1047220DOI10.1080/02331888908802201

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.