Accelerating DAEM Algorithm And It's Application On Computing VaR

Statistical inference helps in perception of the overall population by inferring characteristics of underlying distribution.Parameter estimat.ion is an important part of statistical inference and it is important to have complete data for re-searchers during parameter estimation.However,in reality,we often encounter a large number of missing data.And that's why the hidden variable model has been proposed.Hidden variables,or latent variables,are variables that cannot be direct-ly observed but whose properties can be inferred from observed data.Maximum likelihood estimation is an important method of parameter estimation for hidden variable model.However,when the model is complex,we usually cannot get clear solution of maximum likelihood estimation.Expectation maximization(EM)al-gorithm is an iterative algorithm to find local maximum likelihood parameters when it is impossible to directly solve the equation.Because of its good conver-gence and stability,EM algorithm has been widely concerned by academia and industry.However,EM algorithm also has disadvantages of slow convergence and local optimum.Many scholars have made contributions in improving EM algo-rithm.Deterministic Annealing EM algorithm was proposed in 1998 by Naonori Ueda and Ryohei Nakano[46]in order to overcome the local optimal problem of EM algorithm.A new posterior density of"temperature"is proposed by means of maximum entropy and controlled annealing process.Since the hidden variable model is widely used,the application scenarios of EM algorithm are very rich.Risk measurement is an important application for EM algorithm.In 1997,J.P.Morgan first proposed value-at-risk(VaR)method:VaR?(X)= inf{x ? R:P[X?x]>?} = Fx-1(a)The implication can be simply stated:the probability that the loss will not exceed VaR in the following period of time is a.VaR att,empts to provide a single measure of risk for the portfolio of financial institutions,which can reflect the overall risk of financial institutions.In practical,financial data show obvious characteristic of sharp peak and thick tail,so it is very important to use appropriate distribution to fit it.Zangari[54]proposed to use the Gaussian Mixed Model(GMM)to fit the thick-tailed distribution.Venkataraman[55]presents a method for estimating VaR using a binary mixed gaussian model.Zhang[58]proposed some improve-ment measures.In these improvements,it is undoubtedly necessary to use EM algorithm to estimate and do parameter estimation works.Therefore,the study oin improved EM algorithm is of great significance for quantifying financial risks.In this paper,the Aitken acceleration method is applied to Deterministic An-nealing EM algorithm.A binary mixed normal distribution example verifies that using Aitken method can reduce the cycle number of DAEM algorithm and there-by accelerate DAEM algorithm.However,the DAEM-Aitken algorithm cannot be widely promoted due to the defects of Aitken acceleration method,such as its instability.Finally,this paper applies DAEM algorithm to risk measurement.We use the daily closing price data of Shanghai Composite Index(000001.sh)and Standard&Poor's 500(spx.gi)from 1999 to 2018,and apply DAEM algorithm to fit the distribution of loss data so as to obtain the CVAR and VAR values of Chinese marketand American market.Firstly,VaR value and CVa.R value sat-isfying BIC criterion are calculated.The mean value,variance,probability and other aspects of the Chinese market and the American market under positive and extreme economic conditions are analyzed and compared.The conclusion shows that there are greater risks in the Chinese market.Then,by changing the num-ber of distributions and comparing the financial markets of China and the United States in the three dimensions of mean,variance and probability of occurrence,similar conclusions are obtained.