Research On Some Heavy-tailed Integer-valued Time Series Models

Integer-valued time series are commonly observed in many applied fields such as economics,finance,biology,computer science,electronic engineering,environmental science,medicine,insurance,and so on.A fundamental objective of statistical analysis with count data is to capture relevant features,such as overdispersion or underdisper-sion,zero inflation and even heavy-tailedness.The classical Gaussian ARMA-type pro-cesses can not be capable of capturing the features of integer-valued time series,such as overdispersion,asymmetric marginal distributions,or excess of zeros.Many models for dealing with this type of data have been proposed and studied,where integer-valued au-toregressive(INAR)models and integer-valued generalized autoregressive conditional heteroscedastic(INGARCH)models are most prominent.The INAR model with Pois-son innovations is equi-dispersed,which is not usually suitable in practical applications.Various generalized versions of the INAR(1)model are considered and investigated to account for zero inflation and overdispersion by appropriately varying the innovation-s'distribution.Another way to extend INAR(1)model is considering different types of the thinning operator instead of only modifying the innovations'distribution.As an alternative,the Poisson INGARCH(1,1)model can only accommodate condition-al equidispersed time series of counts.In practice,however,count data can exhibit other features,namely underdispersion and overdspersion.These departures make the Poisson INGARCH model unsuitable or limited for modeling count data.Although many models were proposed for modelling count time series with overdis-persion and zero-inflation,but heavy-tailedness is less considered.Heavy-tailedness,which implies that the tail probabilities are non-negligible or decrease very slowly,is frequently observed in time series.However,ignoring the tail part may lead to a loss of useful informations.The challenge in modeling heavy-tailed data is that one seeks appropriate distributions to take care of the major part of data and the heavy tail.In this thesis,we generalize and extend the traditional INAR and INGARCH models.The main content of this thesis is divided into three parts,details as follows:1.INAR(1)process with heavy-talied innovations.Poisson inverse Gaussian dis-tribution,as an alternative to the negative binomial,which is useful in accommodating data with tails longer than negative binomial and Poisson distribution,has been pro-posed and studied to model count data for many years.Zhu and Joe(2009)indicated that a generalized Poisson inverse Gaussian(GPIG)distribution family as an extension of Poisson inverse Gaussian distribution is able to account for heavy-tailed count data.The generalized Poisson-inverse Gaussian family is very flexible,which includes Pois-son distribution,Poisson-inverse Gaussian distribution,the discrete stable distribution and so on.The model we proposed,a new INAR(1)process with generalized Poisson-inverse Gaussian innovations,is capable of capturing these features.Stationarity and ergodicity of this model are investigated and the marginal mean and variance are pro-vided.Conditional maximum likelihood is used for estimating the parameters,and consistency and asymptotic normality for the estimators are presented.Further,we consider theh-step forecast and define theh-step ahead predicted root mean squared error.In diagnostics,we consider Person residual and the probability integral trans-form.To show that the proposed model has flexibility in modelling count data,we discuss three applications of the GPIG-INAR(1)model.The first example illustrates that the proposed model can take into account count data with excessive zeros,and the second and third examples show that the proposed model has a good performance in modelling heavy-tailed count data.2.A class of max-INAR(1)process with explanatory variables.In some real examples,we encounter that the path increases in one direction,then decreases,and then goes back and forth.If such count data is concerned,the conventional INAR processes with its additive structure can not be able to capture this up-and-down movement tendency.To solve this problem,the max-AR process was extended to the discrete case,called max-INAR(1)process.Note that the autoregressive parameter is assumed to be a constant.The autoregressive parameter may vary with time and it may be random.For example,X_tdenotes the number of unemployed people in theth month,?°X_t-1is the number of unemployed people from the-1th month and c_t stands for the newly unemployed people in theth month.The unemployment ratemay be affected by various factors,such as country or area polity,economic level and so on.To make this process more flexible,we introduce a class of max-INAR processes with explanatory variables,where the autoregressive parameter in the thinning operator depends on explanatory variables.The conditional maximum likelihood estimation method is used to estimate parameters.We also discuss the possible marginal distribution and the extreme value.The proposed models are satisfied extreme value conditions.Some simulation experiments are conducted to study the finite sample performance of the estimators.In the first example,we consider the daily number of stock trades.We apply the proposed models to the wind direction data.Compared with some existing integer-valued time series models,our proposed models show superior performances.3.Generalized Conway-Maxwell-Poisson integer-valued GARCH models.Model selection has always been an issue in practical application.A strategy is to seek a dis-tribution able to capture features of count data.Imoto(2014)indicted that the gener-alized Conway-Maxwell-Poisson distribution generalizes the Conway-Maxwell-Poisson distribution by adding a parameter,which plays the role of controlling the length of the tail.We define a straightforward reparametrization of the GCOM-Poisson distri-bution based on an approximation to the expectation.We propose a more flexible INGARCH model based on the generalized COM-Poisson distribution,which offers a unified framework to deal with underdispersed or overdispersed,zero-inflated and heavy-tailed time series of counts.We investigate basic properties of the proposed model and obtain estimators of parameters via the conditional maximum likelihood method.Some simulation experiments are conducted to study the finite sample per-formance of the estimators for unknown parameters by Monte Carlo simulations.We also fit models to three real data sets,which demonstrates that our proposed model is quite flexible and practical for modeling and inference about time series of counts compared with some existing integer-valued time series models.