Convolutional Auto-Regressive Moving Average Models For Functional Time Series

With the remarkable development of time series analysis,functional time series analysis is becoming attractive to satisfy the complexity of data analysis.The research of predecessor is mainly on the linear process in functional spaces such as FAR(p)model.The definition and some theorems of AR(p)models are generalized to functional spaces.In addition,the parameter estimation uses principle component analysisspline models and kernel smoothing methods as major approach.In this thesis,we generalize CFAR(p)model to a more complex model which is called CFARMA(p,q).The difference between CFARMA and CFAR is the moving average part.In section 3,we give a sufficient condition to the weak stationarity of CFARMA(p,q)model.In the proof of this theorem,we use theory of matrices and functional analysis as well as the lemma given by Bosq(2000).Afterwards,we use MLE method to estimate the parameter of CFARMA(p,q)model,assisted by B-spline basis function.We cannot calculate the stochastic term directly because of the integration of convolutional function.Therefore,we use high order auto-regressive model to fit ?t.We estimate the degrees of freedom on B-spline basis function by penalty function and use AIC principle to determine the order p,q.Finally,the finite sample performance of the proposed estimator for the CFARMA models is discussed through simulation studies.