Robust Empirical Characteristic Function Theory And Application

In finance and industrial production,we always encounter asymmetric data.By further processing and analysing these data,a plenty of data are found to have thick tail,multi-peak and biased properties,and do not obey normal distribution.If we just simply hypothesize data from normal distribution,the results will have a lot of errors.Therefore,it is necessary to find more suitable distribution fitting asymmetric data.Skew normal distribution and the three-parameter inverse Gaussian distribution are generalizations of normal distribution,and they preserve some advantages of normal distribution.For example,it is easy to be used in different ways by simple transformations.At the same time,skew-normal distribution and the three-parameter inverse Gaussian distribution also have high efficiency when dealing with thick tail,multi-peak and biased data.Therefore,they are widely used in asymmetric data processing and analysis.When using these two distributions to fit actual data,we need to estimate their parameters.But it is hard to estimate their parameters,since the profile likelihood of shape parameter can be very flat as it goes to infinity,or the Fisher's information matrixes of their shape parameters are singular.Consequently,the maximum likelihood estimate of shape parameter can be ? and the mle's of other model parameters do not exist.The characteristic function has an advantage of uniform boundedness,so the empirical characteristic function method can also be a parameter estimation method.However,when minimizing its distance function,the results of its often have outliers.Moreover,its estimation efficiency is also affected by the selection of grid points.Therefore,referring to the idea of M-estimation method,a more robust empirical characteristic function method is proposed in Chapter 2,so there is no outlier appearing when the distance function is minimized.We also prove that the estimated value of the new method converges to true values in probability.In the last two chapters,the robust characteristic function method is applied to estimate parameters of skew normal distribution and the threeparameter inverse Gaussian distribution under different sample sizes.According to the simulation results,the estimators of the robust empirical characteristic function method are smaller than that of other measures in term of mean square errors and absolute biases.Therefore,the robust empirical characteristic function can be a new way to estimate parameters of skew normal distribution and the three-parameter inverse Gaussian distribution.