A seasonal geometric INAR process based on negative binomial thinning operator
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A seasonal geometric INAR(1) process based on negative binomial thinning operator Shengqi Tian1 · Dehui Wang1 · Shuai Cui1 Received: 28 November 2017 / Revised: 18 July 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In this article, we propose a new seasonal geometric integer-valued autoregressive process based on the negative binomial thinning operator with seasonal period s. Some basic probabilistic and statistical properties of the model are discussed. Conditional maximum likelihood estimators are obtained, and the asymptotic properties of the estimators are established. Some theoretical results of point forecasts are obtained. Numerical results are presented. At the end, two real data examples are investigated to assess the performance of our new model. Keywords Seasonality · Over-dispersion · Negative binomial thinning operator · Geometric distribution · Estimate · Forecast
1 Introduction In certain situations, it becomes necessary to deal with seasonal non-negative integervalued time series. This kind of time series are encountered in many fields, such as actuarial science, economic, epidemiology, transport and so on. They often exhibit a marked seasonal pattern with period s, when similarities in the series occur after s basic time intervals. Thus, this property is called seasonality, and the basic time interval is seasonal period. Time series with seasonality may stem from factors such as weather or its intrinsic qualities. Over the past few decades, a selection of models based on different schemes are proposed to analyze non-negative integer-valued time series data. Among the various
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Dehui Wang [email protected] Shengqi Tian [email protected] Shuai Cui [email protected]
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School of Mathematics, Jilin University, Changchun 130012, China
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S. Tian et al.
models, the class of models based on thinning scheme has received wide attention in last three decades. In contrast to other integer-valued models, the derivation of statistical properties of these models is relatively straightforward using the concepts of closure under convolution and infinite divisibility of distributions. A prime thinning operator is the binomial thinning operator introduced by Steutel and van Harn (1979). The first-order integer-valued autoregressive (INAR(1)) models based on this operator were proposed by McKenzie (1985) and Al-Osh and Alzaid (1987). With deepening of research, there are many modifications and generalizations in respect of model order, marginal distribution and autoregressive coefficient, see McKenzie (1986), Al-Osh and Alzaid (1988), Alzaid and Al-Osh (1988, 1990, 1993), Du and Li (1991), Al-Osh and Aly (1992), Brännäs (1995), Zhu and Joe (2006), Weiß (2008a), Zheng et al. (2006, 2007), Monteiro et al. (2010) and so on. Nevertheless, the binomial thinning operator which contains Bernoulli counting series has its own limitations. Therefore, some other type of thinning operators are proposed. Motivated by the aim of defining an INAR(1) model for counts with over-dispers
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