Statistical Inferences For Two Classes Of Hysteretic Integer-Valued Autoregressive Models

Integer time series is the counting data formed by a certain phenomenon at different moments within a certain time range.This kind of data widely exists in various fields of production and life.From the value range,the values of these data are non-negative integers and can be divided into two categories.One is the observation data with a limited range,such as the number of days of rain per week in a region and the number of inpatients per day in a regional hospital,etc.The other is an infinite range of observations,such as the number of claims an insurance company makes each month or the number of earthquakes worldwide each year.In this paper,we mainly consider the statistical inference of two kinds of integer value autoregressive models with time delay.The main content of this article is divided into two parts.In the first part,a kind of self-excited hysteretic binomial autoregressive model is proposed.The model follows the binomial distribution with the condition of past information,and presents as a hysteretic autoregressive structure,which is used to describe the integer value data in a limited range.The strict stationarity and ergodicity of the process are proved.Some probabilistic statistical properties and parameter estimation methods of the model are studied,including conditional least squares estimation,weighted conditional least squares estimation and maximum likelihood estimation,and their asymptotic properties are given.The search algorithms of the two boundary parameters and the strong consistency of the corresponding estimators are given.Then the results of the three estimation methods are verified by numerical simulation.Finally,the model was applied to a group of German meningitis data,and the fitting effect of the data was compared under different integer value time series models.The empirical results showed that the model had the best fitting effect.In the second part,we further study the infinite range of integer time series,and propose a kind of self-excited integer lag autoregressive model.The new information sequence of this model is driven by Poisson distribution,which can be used to describe the infinite range of integer data.The stationary ergodic property of the process is proved,and the probabilistic statistic property is given.The parameters are estimated by conditional least squares estimation,weighted conditional least squares estimation and maximum likelihood estimation,and the asymptotic property is given,and a more reasonable boundary parameter search algorithm is given.Finally,the estimated effect is studied by comparing some simulations and applied to an empirical analysis of a set of crime data in Australia.