Robust estimation for general integer-valued time series models
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Robust estimation for general integer-valued time series models Byungsoo Kim1 · Sangyeol Lee2 Received: 17 October 2018 / Revised: 8 April 2019 © The Institute of Statistical Mathematics, Tokyo 2019
Abstract In this study, we consider a robust estimation method for general integer-valued time series models whose conditional distribution belongs to the one-parameter exponential family. As a robust estimator, we employ the minimum density power divergence estimator, and we demonstrate this is strongly consistent and asymptotically normal under certain regularity conditions. A simulation study is carried out to evaluate the performance of the proposed estimator. A real data analysis using the return times of extreme events of the Goldman Sachs Group stock is also provided as an illustration. Keywords Robust estimation · Minimum density power divergence estimator · General integer-valued time series · One-parameter exponential family · INGARCH models
1 Introduction In recent years, integer-valued time series models have received considerable attention from researchers in diverse research areas. Since the work of McKenzie (1985) and Al-Osh and Alzaid (1987), integer-valued autoregressive (INAR) models based on a binomial thinning operator have been widely employed to analyze correlated time series of counts. See Weiß (2008) for a review. Although INAR models are useful in many cases, the equidispersion property that arises in the INAR model with Poisson innovations can lead to a serious problem, because many real datasets exhibit overdispersion. To remedy this, Ferland et al. (2006) proposed to use Poisson integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models, and
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Sangyeol Lee [email protected] Byungsoo Kim [email protected]
1
Department of Statistics, Yeungnam University, Kyungsan 38541, Korea
2
Department of Statistics, Seoul National University, Seoul 08826, Korea
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B. Kim, S. Lee
later, Fokianos et al. (2009) developed Poisson autoregressive (Poisson AR) models, including nonlinear specifications for their intensity processes. These models not only merit keeping the Poisson distribution as their underlying distributions but also capture the over-dispersion phenomenon effectively. Researchers invested considerable efforts to relax the Poisson assumption in INGARCH models and extended the Poisson INGARCH model to other distributional models. Examples include negative binomial INGARCH (NB-INGARCH) models (Davis and Wu 2009; Christou and Fokianos 2014) and zero-inflated generalized Poisson INGARCH models (Zhu 2012a, b; Lee et al. 2016). Davis and Liu (2016) recently considered one-parameter exponential family AR models, called general integer-valued time series models. Diop and Kengne (2017) and Lee and Lee (2018) then utilized this framework to handle the problem of detecting a change point. In these articles, the conditional maximum likelihood estimator (CMLE) is employed for parameter estimation. However, the CMLE is sensitive to outliers and a model bias when an ex
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