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Difference Equations for the Higher Order Moments and Cumulants of the INAR(p) Model

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  • Maria Eduarda Silva
  • Vera Lúcia Oliveira

Abstract

. Here we obtain difference equations for the higher order moments and cumulants of a time series {Xt} satisfying an INAR(p) model. These equations are similar to the difference equations for the higher order moments and cumulants of the bilinear time series model. We obtain the spectral and bispectral density functions for the INAR(p) process in state–space form, thus characterizing it in the frequency domain. We consider a frequency domain method – the Whittle criterion – to estimate the parameters of the INAR(p) model and illustrate it with the series of the number of epilepsy seizures of a patient.

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  • Maria Eduarda Silva & Vera Lúcia Oliveira, 2005. "Difference Equations for the Higher Order Moments and Cumulants of the INAR(p) Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(1), pages 17-36, January.
    • Handle: RePEc:bla:jtsera:v:26:y:2005:i:1:p:17-36
    DOI: 10.1111/j.1467-9892.2005.00388.x
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    References listed on IDEAS

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    1. M. A. Al‐Osh & A. A. Alzaid, 1987. "First‐Order Integer‐Valued Autoregressive (Inar(1)) Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 8(3), pages 261-275, May.
    2. S. A. O. Sesay & T. Subba Rao, 1992. "Frequency‐Domain Estimation Of Bilinear Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 13(6), pages 521-545, November.
    3. Rice, John, 1979. "On the estimation of the parameters of a power spectrum," Journal of Multivariate Analysis, Elsevier, vol. 9(3), pages 378-392, September.
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    1. Feike C. Drost & Ramon Van Den Akker & Bas J. M. Werker, 2008. "Local asymptotic normality and efficient estimation for INAR(p) models," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(5), pages 783-801, September.
    2. Jiayue Zhang & Fukang Zhu & Huaping Chen, 2023. "Two-Threshold-Variable Integer-Valued Autoregressive Model," Mathematics, MDPI, vol. 11(16), pages 1-20, August.
    3. Silva, Isabel & Silva, M. Eduarda, 2006. "Asymptotic distribution of the Yule-Walker estimator for INAR(p) processes," Statistics & Probability Letters, Elsevier, vol. 76(15), pages 1655-1663, September.
    4. 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.
    5. Isabel Silva & M. Eduarda Silva & Isabel Pereira & Nélia Silva, 2005. "Replicated INAR(1) Processes," Methodology and Computing in Applied Probability, Springer, vol. 7(4), pages 517-542, December.

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