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Geometric infinitely divisible autoregressive models

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Abstract

In this article, we discuss some geometric infinitely divisible (gid) random variables using the Laplace exponents which are Bernstein functions and study their properties. The distributional properties and limiting behavior of the probability densities of these gid random variables at \(0^{+}\) are studied. The autoregressive (AR) models with gid marginals are introduced. Further, the first order AR process is generalized to kth order AR process. We also provide the parameter estimation method based on conditional least square and method of moments for the introduced AR(1) process. We also apply the introduced AR(1) model with geometric inverse Gaussian marginals on the household energy usage data which provide a good fit as compared to normal AR(1) data.

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Notes

  1. https://www.kaggle.com/datasets/jaganadhg/house-hold-energy-data.

References

  • Grunwald GK, Adrian ER, Guttorp P (1993) Time series of continuous proportions. Journal of Royal Statistical Society. Series B (Methodological) 55(1):103–116

  • Gaver D, Lewis P (1980) First-order autoregressive gamma sequences and point processes. Advances in Applied Probability 12–3:727–745

    Article  MathSciNet  Google Scholar 

  • Dewald LS, Lewis PAW (1985) A new Laplace second-order autoregressive time series model. IEEE Transactions on Information Theory 31–5:645–651

    Article  MathSciNet  Google Scholar 

  • Abraham B, Balakrishna N (1999) Inverse Gaussian autoregressive models. Journal of Time Series Analysis 20–6:605–618

    Article  MathSciNet  Google Scholar 

  • Bhootna N, Kumar A (2022) Tempered stable autoregressive models. Communications in Statistics-Theory and Methods, 1–21

  • Lekshmi VS, Jose KK (2004) An autoregressive process with geometric \(\alpha -\) Laplace marginals. Statistical Papers 45:337–350

    Article  MathSciNet  Google Scholar 

  • Lekshmi VS, Jose KK (2006) Autoregressive processes with Pakes and geometric Pakes generalized Linnik marginals. Statistics and Probability Letters 76:318–326

    Article  MathSciNet  Google Scholar 

  • Grunwald G, Hyndman R, Tedesco L (1996) A unified view of linear ar(1) models. D.P. Monash University, Clayton

    Google Scholar 

  • Kleinbaum GD, Klein M (2012) Survival Analysis: Self-Learning Text. Springer, New York, NY

    Book  Google Scholar 

  • Applebaum D (2009) Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (UK)

    Book  Google Scholar 

  • Kumar A, Nane E (2018) On the infinite divisibility of distributions of some inverse subordinators. Modern Stochastic: Theory and Applications 5:509–519

    MathSciNet  Google Scholar 

  • Kumar A, Vellaisamy P (2015) Inverse tempered stable subordinators. Statistics and Probability Letters 103:134–141

    Article  MathSciNet  Google Scholar 

  • Schilling RL, Song R, Vondracek Z (eds) (2012) Bernstein Functions: Theory and Applications. De Gruyter, Berlin, Boston

    Google Scholar 

  • Klebanov BL, Maniya GM, Melamed IA (1985) A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables. Theory of Probability & Its Applications 29(4):791–794

    Article  Google Scholar 

  • Feller W (1971) An Introduction to Probability Theory and Its Applications. Wiley, New York

    Google Scholar 

  • Ridout MS (2009) Generating random numbers from a distribution specified by its Laplace transform. Statistics and Computing 19(4):439–450

    Article  MathSciNet  Google Scholar 

  • Schiff J (1999) The Laplace Transform: Theory and Applications. Springer, New York

    Book  Google Scholar 

  • Abramowitz M, Stegun IA (1954) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Conference on Mathematical Tables, Cambridge (UK)

Download references

Acknowledgements

Monika S. Dhull would like to thank the Ministry of Education (MoE), India for supporting her PhD research. Further, Arun Kumar would like to express his gratitude to Science and Engineering Research Board (SERB), India for financial support under the MATRICS research grant MTR/2019/000286.

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  1. Monika S. Dhull and Arun Kumar have contributed equally to this work.

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  1. Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab, 140001, India

    Monika S. Dhull & Arun Kumar

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  1. Monika S. Dhull
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  2. Arun Kumar
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All authors contributed equally.

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Correspondence to Arun Kumar.

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Dhull, M.S., Kumar, A. Geometric infinitely divisible autoregressive models. Stat Papers (2024). https://doi.org/10.1007/s00362-024-01564-y

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  • DOI: https://doi.org/10.1007/s00362-024-01564-y

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