{"title":"The Non-Stationary BINARMA(1,1) Process with Poisson Innovations: An Application on Accident Data","authors":"Y. Sunecher, N. Mamode Khan, V. Jowaheer","volume":155,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":203,"pagesEnd":207,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10010899","abstract":"This paper considers the modelling of a non-stationary
\r\nbivariate integer-valued autoregressive moving average of order
\r\none (BINARMA(1,1)) with correlated Poisson innovations. The
\r\nBINARMA(1,1) model is specified using the binomial thinning
\r\noperator and by assuming that the cross-correlation between the
\r\ntwo series is induced by the innovation terms only. Based on
\r\nthese assumptions, the non-stationary marginal and joint moments
\r\nof the BINARMA(1,1) are derived iteratively by using some initial
\r\nstationary moments. As regards to the estimation of parameters of
\r\nthe proposed model, the conditional maximum likelihood (CML)
\r\nestimation method is derived based on thinning and convolution
\r\nproperties. The forecasting equations of the BINARMA(1,1) model
\r\nare also derived. A simulation study is also proposed where
\r\nBINARMA(1,1) count data are generated using a multivariate
\r\nPoisson R code for the innovation terms. The performance of
\r\nthe BINARMA(1,1) model is then assessed through a simulation
\r\nexperiment and the mean estimates of the model parameters obtained
\r\nare all efficient, based on their standard errors. The proposed model
\r\nis then used to analyse a real-life accident data on the motorway in
\r\nMauritius, based on some covariates: policemen, daily patrol, speed
\r\ncameras, traffic lights and roundabouts. The BINARMA(1,1) model
\r\nis applied on the accident data and the CML estimates clearly indicate
\r\na significant impact of the covariates on the number of accidents on
\r\nthe motorway in Mauritius. The forecasting equations also provide
\r\nreliable one-step ahead forecasts.","references":" E. McKenzie, \u201cSome ARMA models for dependent sequences of Poisson\r\ncounts,\u201d Advances in Applied Probability, vol. 20, pp. 822\u2013835, 1988.\r\n M. Al Osh and A. Alzaid, \u201cFirst-order integer-valued autoregressive\r\nprocess,\u201d Journal of Time Series Analysis, vol. 8, pp. 261\u2013275, 1987.\r\n K. Brannas, \u201cExplanatory variable in the AR(1) count data model,\u201d\r\nUmea University, Department of Economics, vol. No.381, pp. 1\u201321,\r\n1995.\r\n V. Jowaheer and B. Sutradhar, \u201cFitting lower order nonstationary\r\nautocorrelation models to the time series of Poisson counts,\u201d\r\nTransactions on Mathematics, vol. 4, pp. 427\u2013434, 2005.\r\n N. Mamode Khan and V. Jowaheer, \u201cComparing joint GQL estimation\r\nand GMM adaptive estimation in COM-Poisson longitudinal regression\r\nmodel,\u201d Commun Stat-Simul C., vol. 42(4), pp. 755\u2013770, 2013.\r\n K. Brannas and A. Quoreshi, \u201cInteger-valued moving average modelling\r\nof the number of transactions in stocks,\u201d Applied Financial Economics,\r\nvol. No.20(18), pp. 1429\u20131440, 2010.\r\n M. Al Osh and A. Alzaid, \u201cInteger-valued moving average (INMA)\r\nprocess,\u201d Statistical Papers, vol. 29, pp. 281\u2013300, 1988a.\r\n A. Nastic, P. Laketa, and M. Ristic, \u201cRandom environment\r\ninteger-valued autoregressive process,\u201d Journal of Time Series Analysis,\r\nvol. 37(2), pp. 267\u2013287, 2016.\r\n X. Pedeli and D. Karlis, \u201cBivariate INAR(1) models,\u201d Athens University\r\nof Economics, Tech. Rep., 2009.\r\n X. Pedeli and D.Karlis, \u201cSome properties of multivariate INAR(1)\r\nprocesses.\u201d Computational Statistics and Data Analysis, vol. 67, pp.\r\n213\u2013225, 2013.\r\n P. Popovic, M. Ristic, and A. Nastic, \u201cA geometric bivariate time series\r\nwith different marginal parameters,\u201d Statistical Papers, vol. 57, pp.\r\n731\u2013753, 2016.\r\n M. Ristic, A. Nastic, K. Jayakumar, and H. Bakouch, \u201cA bivariate\r\nINAR(1) time series model with geometric marginals,\u201d Applied\r\nMathematical Letters, vol. 25(3), pp. 481\u2013485, 2012.\r\n Y. Sunecher, N. Mamodekhan, and V. Jowaheer, \u201cA gql estimation\r\napproach for analysing non-stationary over-dispersed BINAR(1) time\r\nseries,\u201d Journal of Statistical Computation and Simulation, 2017.\r\n A. Quoreshi, \u201cBivariate time series modeling of financial count\r\ndata,\u201d Communication in Statistics-Theory and Methods, vol. 35, pp.\r\n1343\u20131358, 2006.\r\n A.M.M.S.Quoreshi, \u201cA vector integer-valued moving average model for\r\nhigh frequency financial count data,\u201d Economics Letters, vol. 101, pp.\r\n258\u2013261, 2008.\r\n Y. Sunecher, N. Mamodekhan, and V. Jowaheer, \u201cEstimating the\r\nparameters of a BINMA Poisson model for a non-stationary bivariate\r\ntime series,\u201d Communication in Statistics: Simulation and Computation,\r\nvol. [Accepted for Publication on 27 June 2016], 2016.\r\n C. Weib, M. Feld, N. Mamodekhan, and Y. Sunecher, \u201cInarma modelling\r\nof count series,\u201d Stats, vol. 2, pp. 289\u2013320, 2019.\r\n S. Kocherlakota and K. Kocherlakota, \u201cRegression in the bivariate\r\nPoisson distribution,\u201d Communications in Statistics-Theory and\r\nMethods, vol. 30(5), pp. 815\u2013825, 2001.\r\n F. Steutel and K. Van Harn, \u201cDiscrete analogues of self-decomposability\r\nand statibility,\u201d The Annals of Probability, vol. 7, pp. 3893\u2013899, 1979.\r\n I. Yahav and G. Shmueli, \u201cOn generating multivariate Poisson data\r\nin management science applications,\u201d Applied Stochastic Models in\r\nBusiness and Industry, vol. 28(1), pp. 91\u2013102, 2011.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 155, 2019"}