Parametric Poisson Bifurcated Autoregressive Process: Application to Worldwide, Regional, and Peculiar Countries’ of Automobile Production


  • Rasaki Olawale Olanrewaju Africa Business School, Mohammed VI Polytechnic University
  • Sodiq Adejare Olanrewaju Department of Statistics, University of Ibadan, Ibadan, 900001, Nigeria.
  • Toyin Omoyeni Oguntola Department of Statistics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria.
  • Wasiu Adepoju Department of Mathematics Education, University of Ibadan, Ibadan, Nigeria.


Automobile Production, Lindeberg’s Condition, Martingale, Poisson Bifurcated Autoregressive (PBAR), Weighted Least Squares (WLS).


This article introduces Bifurcated Autoregressive (BAR) process with two apart marginal distribution error terms of  w2 and w2+1 of Poisson white noises to make it Poisson Bifurcated Autoregressive (PBAR) in a parametric setting. The statistical definition of PBAR (1) process with parameters B1 and B2 that must be |B1 | and |B2 |<1 for stationary process was spelt-out. Weighted Least Squares (WLS) parameter estimation technique was adopted and the process limiting distribution was carried-out via the combination methods of martingale process and Lindeberg’s condition. Monthly automobile production in Japan, Outside Japan, America, USA, Europe, Asia, and China that approximately tantamount to worldwide, regional, and peculiar countries’ of automobile production was subjected to the PBAR process. In conclusion, Japan automobile production possessed the highest and largest error correlation (w2 , w2+1 ) of 0.6582 (65%) with first order PBAR, with B1Y(t/2) , such that B1=0.2228 of degenerated two major divisions of automobile production of Registrations and Mini-Vehicles with descendant of different brands (models).


Download data is not yet available.


Al-Osh, M.A. & Alzaid, A.A. (1987). First-order integer-valued Autoregressive (INAR (1))

process. Journal of Time Series. Analysis, Vol. 8(3), pp. 261–275.

Bercu, B., Saporta, B.D., & Petit, A.G. (2008). Asymptotic Analysis for Bifurcating

Autoregressive Processes via a Martingale Approach. ArXiv:0807.0528v1.

Boshnakov, G.N. (2006). Prediction with Mixture Autoregressive Models. Research Report No.

, 2006, Probability and Statistics Group School of Mathematics, The University of

Manchester, London.

Cowan, R. & Staudte, R.G. (1986). The Bifurcating Autoregressive Model in Cell Lineage

Studies. Biometrics, Vol. 42, 769–783.

Cowan, R. (1984). Statistical Concepts in the Analysis of Cell Lineage Data. 1983 Workshop

Cell Growth Division (pp. 18-22). Melbourne: Latrobe University.

Elbayoumi, T.M. E. (2017). A robust estimate for the bifurcating autoregressive model with

application to cell lineage data. Western Michigan University Scholar Works at WMU.

Haccou, P., Jagers P. & Vatutin, V. (2005). Branching Processes: Variation, Growth, and

Extinction of Populations. Cambridge: Cambridge University Press.

Huggins, R.M. & Basawa, I.V. (1999). Extensions of the bifurcating Autoregressive model for

cell lineage studies. Journal of Applied Probability, Vol. 36, pp. 1225–1233.

Huggins, R.M. & Basawa, I.V. (2000). Inference for the Extended Bifurcating Autoregressive

Model for Cell Lineage Studies. Australia New Zealand Journal Statistics, Vol. 42,


Hwang, S.Y. & Basawa, I.V. (2011). Asymptotic optimal inference for Multivariate branching

Markov processes via martingale estimating Functions and mixed normality. Journal of

Multivariate Analysis, Vol. 102, pp. 1018–1031.

Hwang, S.Y. & Choi, M.S. (2011). Preliminary Identification of Branching-Heteroscedasticity

for Tree-Indexed Autoregressive Processes. Communications of the Korean Statistical

Society, Vol. 18(6), pp. 809–816. doi:

Olanrewaju, R.O., Ojo, J.F. & Adekola, L.O. (2020). Bayesian latent Autoregressive Stochastic

Volatility: An Application of Naira to Eleven Exchangeable Currencies Rates. Open

Journal of Mathematical Sciences, Vol. 4(1), pp. 386-396. doi.10.30538/oms2020.0128.

Ojo, J.F. & Olanrewaju, R.O. (2016). On Mixture Autoregressive (MAR) using Naira-Dollar

Exchange Rates. Journal of Nigeria Association Mathematical Physics, Vol. 38(12), pp.


Olanrewaju, R.O. & Olanrewaju, S.A. (2021): An Alternative Mean Variance Portfolio

Theoretical Framework: Nigeria Banks’ Market Shares Analysis. Global Journal of

Business, Economics, and Management. Vol. 11(3), pp. 220-234.


Olanrewaju, R.O. Ojo, J.F. & Adekola, L.O. (2021). Interswitching of Transmuted Gamma

Autoregressive Random Processes. Journal of Mathematics and Statistical Science (ISSN

-2518, USA), Vol.7 (7), 183-202.

Olanrewaju, R.O. & Oseni, E. (2021). GARCH and its Variants’ Model: An Application of

Crude Oil Distributions in Nigeria. International Journal of Accounting Finance and Risk

Management, Vol. 6(1), pp.25-35. doi.10.11648/j.ijafrm.20210601.14.

Ojo, J.F. & Olanrewaju, R.O. (2017). Empirical Distribution and Modelling of Spot Prices of

Nigeria Crude Oil. Journal of Sciences and Multidisciplinary Research, Vol. 9(2),

ISSN:2277-0135, pp. 1-12.

Olanrewaju, R.O. (2018). Integer-Valued Time Series Model via Generalized Linear Models

Technique of Estimation. Intentional Annals of Science, Vol.4 (1), pp.35-43. ISSN: 2456-

Ojo, J.F. & Olanrewaju, R.O. & Folorunsho, S.A (2017). Performance of all Nigeria Banks’

Shares using Student-t Mixture Autoregressive Model. Journal of Engineering and Applied

Scientific Research, Vol. 9(1), ISSN: 2384-6569, pp. 69-82.

Olanrewaju, R.O. & Folorunsho, S.A. (2018). Generalized Autoregressive Score (GAS)

Functions under Gaussian and Student-t Distributions. International Journal of Statistics and

Applied Mathematics, Vol. 3(5), ISSN: 2456-1452, pp. 56-61.

Olanrewaju, R.O., Waititu, A.G., & Nafiu, L.A. (2022): Kullback-Leibler Divergence of Mixture

Autoregressive Random Processes via Extreme-Value-Distributions (EVDs) Noise with

Application of the Processes to Climate Change. Transactions on Machine Learning and

Artificial Intelligence, Vol. 10(1), pp. 1-18. doi:10.14738/tmlai.101.11544.

Olanrewaju, R.O., Waititu, A.G., & Nafiu, L.A. (2021). Bull and Bear Dynamics of the Nigeria

Stock Returns Transitory via Mingled Autoregressive Random Processes. Open Journal of

Statistics. Vol. 11, 870-885. doi:10.4236/ojs.2021.115051.

Olanrewaju, R.O., Barry, T.S., Muse, A.H., & Habineza, A. (2021): Ornstein-Uhlenbeck Process

via Conflated Drive of Brownian motion and L´evy Process and its Application.

Mathematical Theory and Modelling, Vol. 11(3), pp. 12 - 20.

Olanrewaju, R.O., Oseni, E., Adekola, L.O. & Oyinloye, A.A. (2018). On Skew Generalized

Extreme Value-ARMA model: An Application to Average Monthly Temperature

(1901-2016) in Nigeria. International Journal of Statistics and Applied Mathematics,

Vol. 3(1), ISSN: 2456-1452, pp. 20-27.

Powell, E. O. (1955). Some features of the generation times of individual Bacteria. Biometrika,

Vol. 42, pp. 16-44.

Staudte, R.G., Zhang, J., Huggins, R.M. & Cowan, R. (1996). A re- Examination of the Cell

Lineage data of E.O. Powell. Biometrics, Vol. 52, 1214–1222.

Saporta, B.D., Petit, A.G. & Marsalle, L. (2018). Statistical study of asymmetry in cell lineage

data. arXiv:1205.4840v3.

Saporta, B.D., Petit, A.G. & Marsalle, L. (2009). Parameters Estimation for Asymmetric

Bifurcating Autoregressive Processes with Missing data. Electronic Journal of Statistics,

Vol. 12, 589–601. ISSN: 1935-7524.

Wong, C. S. & Li, W. K. (2000). On a Mixture Autoregressive Model. Journal of

Royal Statistical Society, Series B, Statistical. Methodology, Vol. 62(1), pp. 95--115.

Wong, C. S. (1998): Statistical inference for some nonlinear time series models, Ph.D. thesis,

University of Hong Kong, Hong Kong.

Zhou, J. & Basawa, I.V. (2005). Maximum Likelihood Estimation for a First-Order Bifurcating

Autoregressive Process with Exponential Errors. Journal of Time Series Analysis, Vol.26 (6),

pp. 825-842.



How to Cite

Olanrewaju, R. O., Olanrewaju, S. A., Oguntola, . T. O., & Adepoju, W. (2023). Parametric Poisson Bifurcated Autoregressive Process: Application to Worldwide, Regional, and Peculiar Countries’ of Automobile Production. Journal of Mathematical Analysis and Modeling, 4(1), 17–35.