Poisson Lognormal Integer-valued GARCH Model

In recent years, the theory of time series has been perfect, and it has been widely used in psychology, financial economy, signal processing, and so on.Commonly used time series models are autoregressive model, moving average model, autoregressive moving average model and generalized autoregressive conditional heteroscedasticity model.For analyzing the count data, traditional research focus on integrating a mixing prior into mixed Poisson distribution to improve performances. Mixed Poisson distributions are widely used in various applications of count data, to deal with extra-Poisson variation(If the sample variance of the Poisson distribution is larger than the estimated variance)and zero-inflated counts. The zero-inflated Poisson(ZIP) distribution and the zero-inflated negative binomial(ZINB) distribution are as solutions to modelling such data, but neither is likely to handle the above two characteristics at the same time in practical applications. Thus, we propose a novel modle――Poisson lognormal integervalued GARCH model.We first give the definition of the Poisson lognormal distribution, utilizing the lognormal as a mixing prior and introduce the basic nature of Poisson lognormal distribution including the mean, variance and skewness, kurtosis and so on, we next estimate its parameters. Based on this distribution, we give the definition of the Poisson lognormal integer-valued GARCH model, the mean and variance, and we use the maximum likelihood method to estimate parameters. We apply the transform relations of distribution to generating the random samples that follows the newly proposed model. And finally we give the numerical simulation to assess the performance of the estimators, it shows that this model has good performance.