A new mixed first-order integer-valued autoregressive process with Poisson innovations
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A new mixed first‑order integer‑valued autoregressive process with Poisson innovations Daniel L. R. Orozco1 · Lucas O. F. Sales1 · Luz M. Z. Fernández2 · André L. S. Pinho2 Received: 6 May 2020 / Accepted: 29 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Binomial thinning · Forecasting · INARCH(1) process · INAR(1) process · Poisson thinning · Time series
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s1018 2-020-00381-6) contains supplementary material, which is available to authorized users. * Luz M. Z. Fernández [email protected] Extended author information available on the last page of the article
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1 Introduction Many time-series counting processes have been studied recently, since there are events in different contexts that can be measured sequentially over time. Examples are the number of accidents in a company that occur monthly, the number of patients treated in a hospital for an hour, the number of children who die each month from a disease, and the number of days it rained in a particular place in the winter period in a particular zone. In these studies, the intention is not only to explain the past, but also to predict the future. This is where time series becomes important. In this paper, we are interested in a special class of models, called integer-valued autoregressive processes (INAR). The pioneers in dealing with first-order INAR pro
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