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Asymptotic properties of CLS estimators in the Poisson AR(1) model

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  • Keith Freeland, R.
  • McCabe, Brendan

Abstract

Many papers have been written on count valued ARMA models, since they were introduced by Al-Osh and Alzaid [1987. J. Time Ser. Anal. 8, 261-275] and McKenzie [1988. Adv. Appl. Probab. 20, 822-835]. However surprisingly little has been written about estimation of these models and even less about the asymptotic properties of the parameter estimates. In fact, some of the asymptotic properties that do appear and are cited in the literature are incorrect. In this paper we derive a corrected explicit expression for the asymptotic variance matrix of the conditional least squares estimators (CLS) of the Poisson AR(1) process. We also show that the distribution of the CLS estimators is asymptotically equivalent to that of estimators based on the Yule-Walker equations and thus neither is more efficient than the other to this order.

Suggested Citation

  • Keith Freeland, R. & McCabe, Brendan, 2005. "Asymptotic properties of CLS estimators in the Poisson AR(1) model," Statistics & Probability Letters, Elsevier, vol. 73(2), pages 147-153, June.
    • Handle: RePEc:eee:stapro:v:73:y:2005:i:2:p:147-153
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    Citations

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    Cited by:

    1. Feike C. Drost & Ramon van den Akker & Bas J. M. Werker, 2009. "Efficient estimation of auto‐regression parameters and innovation distributions for semiparametric integer‐valued AR(p) models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 467-485, April.
    2. Zeng, Xiaoqiang & Kakizawa, Yoshihide, 2024. "Two-step conditional least squares estimation in ADCINAR(1) process, revisited," Statistics & Probability Letters, Elsevier, vol. 206(C).
    3. Christian Weiß, 2008. "Thinning operations for modeling time series of counts—a survey," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 92(3), pages 319-341, August.
    4. Boris Aleksandrov & Christian H. Weiß, 2020. "Testing the dispersion structure of count time series using Pearson residuals," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(3), pages 325-361, September.
    5. M. Kachour & J. F. Yao, 2009. "First‐order rounded integer‐valued autoregressive (RINAR(1)) process," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(4), pages 417-448, July.
    6. Mohammadipour, Maryam & Boylan, John E., 2012. "Forecast horizon aggregation in integer autoregressive moving average (INARMA) models," Omega, Elsevier, vol. 40(6), pages 703-712.
    7. Miroslav M. Ristić & Aleksandar S. Nastić & Ana V. Miletić Ilić, 2013. "A geometric time series model with dependent Bernoulli counting series," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(4), pages 466-476, July.
    8. Weiß, Christian H. & Schweer, Sebastian, 2016. "Bias corrections for moment estimators in Poisson INAR(1) and INARCH(1) processes," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 124-130.
    9. Jonas Andersson & Dimitris Karlis, 2010. "Treating missing values in INAR(1) models: An application to syndromic surveillance data," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(1), pages 12-19, January.
    10. Subhankar Chattopadhyay & Raju Maiti & Samarjit Das & Atanu Biswas, 2022. "Change‐point analysis through integer‐valued autoregressive process with application to some COVID‐19 data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(1), pages 4-34, February.
    11. Yao Rao & David Harris & Brendan McCabe, 2022. "A semi‐parametric integer‐valued autoregressive model with covariates," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(3), pages 495-516, June.
    12. Wagner Barreto-Souza & Sokol Ndreca & Rodrigo B. Silva & Roger W. C. Silva, 2023. "Non-linear INAR(1) processes under an alternative geometric thinning operator," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 695-725, June.
    13. Zeng, Xiaoqiang & Kakizawa, Yoshihide, 2022. "Bias-correction of some estimators in the INAR(1) process," Statistics & Probability Letters, Elsevier, vol. 187(C).
    14. Christian H. Weiß, 2011. "Detecting mean increases in Poisson INAR(1) processes with EWMA control charts," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(2), pages 383-398, September.
    15. Wagner Barreto-Souza, 2019. "Mixed Poisson INAR(1) processes," Statistical Papers, Springer, vol. 60(6), pages 2119-2139, December.

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