| Count time-series data widely exists in practice.For example,the daily number of transaction in stock market,the number of busy lines in a telephone network noted every hour,the number of road accidents in a town in successive months,the number of people suffering from a specific disease,the number of daily views of a web page,etc.Cox(1981)classified the models used in modeling time series of counts into parameter-driven model and observation-driven model.At present,there are mainly two kinds of models for the latter.One is the first-order integer-value autoregressive model based on thinning operators proposed by Alzaid and Al-Osh(1987),which has been widely con-cerned and studied by researchers.The other is the Poisson integer-value generalized autoregressive conditional heteroscedasticity(INGARCH)model proposed by Ferland et al.(2006).Many authers have studied this model,such as fokianos et al.(2009),Neumann(2011)and doukhan et al.(2012),among others.In addition,in recent years,researchers have invested considerable efforts to relax the Poisson assumption in INGARCH models and extended the Poisson INGARCH model to other distribution-al models.Examples include negative binomial INGARCH(NB-INGARCH)models,generalized Poisson INGARCH models,zero-inflated INGARCH models,COM-Poisson INGARCH model,infinitely divisible INGARCH model,one-parameter exponential family INGARCH model and so on.Maximum likelihood estimation is the commonly used parameter estimation method,but it is highly sensitive to the presence of outliers.A common idea to deal with this problem is using robust estimation.Due to the flexibility of the NB-INGARCH model and the universality of the single-parameter exponential family INGARCH model rel-ative to the Poisson INGARCH model,in this paper,we consider two different robust estimation methods of the parameters for these two models,and the details are as follows:1.Mallows quasi-likelihood estimation for the negative binomial integer-valued GARCH model.This method is proposed by Cantoni and ronchetti(2001).It achieves robustness through limiting the deviations of the observations from the time series and controlling the deviations in the design space by downweighting the leverage points.The commonly used weight functions are Huber and Tukey functions.The tuning constant included in weight function controls the trade-off between robustness and ef-ficiency,therefore,the selection of tuning constant is very important,this paper gives a detailed explanation of this problem.In addition,we establish the consistency and asymptotic normality of the Mallows quasi likelihood estimator,and study the per-formance of Mallows quasi likelihood estimator under different intervention effects by numerical simulation.Finally,we use two practical examples to illustrate the per-formance of Mallows quasi likelihood estimator by in-sample prediction and out-of-sample prediction.2.Minimum density power divergence estimator for the negative binomial integer-valued GARCH models.This procedure is proposed by Basu et al.(1998),which is mainly used to solve the parameter estimation problem of the density distribution.It estimates parameters by minimizing the divergence between the assumed model density and the real density of the potential data.The divergence is essentially a measure,and is loosely referred to as distance.It is also through the tuning constant to control the trade-off between robustness and efficiency,and the relevant literature show that the robust estimator based on the minimum density power divergence has good robustness.Under some assumptions,we prove the existence,uniqueness,consistency and asymptotic normality of the robust estimator.The detailed comparison simulation study and practical examples show the superiority of the robust estimator.3.Mallows quasi-likelihood estimation for the single-parameter exponential family INGARCH model based on modified Tukey biweight function.This work is the exten-sion of the first work aforementioned.The weight function in the robust estimation equation is the derivative function of the modified Tukey’s biweight loss function pro-posed by Li(2018)in the third chapter,the loss function is three times continuous and differentiable,thus we can prove the existence,uniqueness,consistency and asymptotic normality of the Mallows quasi-likelihood estimator by verifying the conditions of lem-ma 1 in Jensen and rahbek(2004)so as to remedy the defect that the related literature can not establish the uniqueness of the similar robust estimator.Simulation and real data examples all demonstrate the excellence of Mallows quasi-likelihood estimator. |
