Abstract
Integer-valued time series, seen as a collection of observations measured sequentially over time, have been studied with deep notoriety in recent years, with applications and new proposals of autoregressive models that broaden the field of study. This work proposes a new mixed integer-valued first-order autoregressive model with Poisson innovations, denoted POMINAR(1), mixing two operators known as binomial thinning and Poisson thinning. The proposed process presents some advantages in relation to the most common Poisson innovation processes: (1) this new process allows to capture structural changes in the data; (2) if there are no structural changes, the most common processes with Poisson innovations are particular cases of POMINAR(1). Another important contribution of this work is the establishment of the POMINAR(1) theoretical results, such as the marginal expectation, marginal variance, conditional expectation, conditional variance, transition probabilities. Moreover, the Conditional Maximum Likelihood (CML) and Yule-Walker (YW) estimators for the process parameters are studied. We also present three techniques for one-step-ahead forecasting, the nearest integer of the conditional expectation, conditional median and mode. A simulation study of the forecasting procedures, considering the two estimators, CML and YW methods, is performed, and prediction intervals are presented. Finally, we show an application of the proposed process to a real dataset, referred here as larceny data, including a residual analysis.
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Notes
The following symbol \({\mathop {=}\limits ^{\blacktriangle }}\) means by definition.
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This study was financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)—Brazil—Finance Code 001.
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The main contribution of this work is to propose a new mixed first-order autoregressive model with Poisson innovations, denoted POMINAR(1), mixing two operators known as binomial thinning and Poisson thinning. The proposed process presents some advantages in relation to the most common Poisson innovation processes: (i) This new process allows to capture structural changes in the data; (ii) if there are no structural changes, the most common processes with Poisson innovations are particular cases of POMINAR(1). Another important contribution of this work is the establishment of the POMINAR(1) theoretical results, such as the marginal expectation; marginal variance; conditional expectation; conditional variance; and transition probabilities. Moreover, the Conditional Maximum Likelihood (CML) and Yule-Walker (YW) estimators for the process parameters are studied, and an application to a real dataset is given, showing the effectiveness of the proposed model. Finally, we study three techniques for one-step-ahead forecasting: the nearest integer of the conditional expectation, conditional median and mode. All three techniques are evaluated considering two estimators: CML and YW methods, and their prediction intervals are established. A simulation study of the forecasting procedures is presented, and an application to real data is shown with a residual analysis.
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Orozco, D.L.R., Sales, L.O.F., Fernández, L.M.Z. et al. A new mixed first-order integer-valued autoregressive process with Poisson innovations. AStA Adv Stat Anal 105, 559–580 (2021). https://doi.org/10.1007/s10182-020-00381-6
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DOI: https://doi.org/10.1007/s10182-020-00381-6