| With the development of information technology,more and more attention has been paid to financial high-frequency data analysis.The autoregressive conditional duration(ACD)model plays an important role in dealing with the positive time series of financial high-frequency data.In the general case,the error of the duration model is dense in the vicinity of 0,and presents an asymmetric heavy-tailed distribution,so the distribution of the heavy-tailed shape is more and more used by the statisticians in the ACD model.In fact,setting the exponential distribution,Weibull distribution and generalized gamma distribution as an error distribution for ACD model has been able to handle most of the high frequency data.However,there is a common disadvantage of these models,that is,the fitting effect of the model error sequence is not very good,and the estimation of the model parameters can only be quasi maximum likelihood estimation.As the amount of data is increasing more and more,data structure becomes more and more complex,so we still need to seek a new error distribution function which is more flexible to fitting duration model.It has been proved theoretically that the distribution of mixed Erlang distribution is dense in the positive continuous type distribution and can be approximated by any finite mixed Erlang distribution.Moreover,in estimating the ACD model,the sample size is thousands of,which is entirely suitable for the use of mixed Erlang distribution.We consider a data model in the insurance loss distribution widely used which is called Mixed Erlang distribution(MER distribution)as a new error distribution model of ACD,and learn the modeling idea that introducing latent variables to construct the complete data log likelihood function in order to fitting calculation.In this paper,on the basis of CMM-GEM algorithm,using extended algorithm from EM algorithm which called ECM algorithm,thus using a new algorithm which called CMM-GECM algorithm is performed for parameter fitting by using maximum likelihood estimation,first to obtain the expectations of complete data log likelihood function in the E-step,then in the CM-step the parameters set ψ is divided into ψACD,ψMER two sets of parameters,and one fixed on another set of parameters for conditional maximum,next calculate the log likelihood function of observed data for the standardized model,to determine whether they meet the stop condition,finally we get the results of fitting parameters of MER-ACD model.We use R software to design CMM-GECM algorithm,fitting MER-ACD model respectively to simulated data and financial data,and compared with the Exponential distribution,Weibull distribution and generalized Gamma distribution duration model,to prove the effectiveness of the CMM-GECM algorithm,and the MER-ACD model is superior to the existing ACD model in the financial high-frequency data analysis. |
