A new mixed first-order integer-valued autoregressive process with Poisson innovations

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-o...

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Bibliographic Details
Published in:Advances in statistical analysis : AStA : a journal of the German Statistical Society 2021-12, Vol.105 (4), p.559-580
Main Authors: Orozco, Daniel L. R., Sales, Lucas O. F., Fernández, Luz M. Z., Pinho, André L. S.
Format: Article
Language:English
Subjects:
Autoregressive models
Autoregressive processes
Econometrics
Economics
Estimators
Finance
Forecasting
Innovations
Insurance
Management
Mathematics and Statistics
Mixed integer
Original Paper
Probability Theory and Stochastic Processes
Process parameters
Series (mathematics)
Statistics
Statistics for Business
Thinning
Transition probabilities
Variance
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Summary: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.
ISSN:1863-8171
1863-818X
DOI:10.1007/s10182-020-00381-6