| Symmetric probability statistical distribution such as normal distribution,t distribution and double exponential distribution has been widely used in various fields of society.Since Azzalini proposed asymmetric statistical distribution such as skew normal distribution and skew-t distribution,the skew normal distribution and skew-t distribution have also been applied to various fields in practical life.In practical application,the distribution of the data types are often unknown,choose what kind of distribution to deal with data is one of the important concern of academic and application field,this thesis try to skew normal distribution,skew-t distribution,normal distribution,t distribution study,analyzing the distribution of the selection and application in the actual data,the main work is as follows.Firstly,Monte Carlo method is used to generate simulated data of different distributions,and the maximum likelihood estimation statistics of the population mean and variance of the normal distribution,skew normal distribution,t distribution and skew-t distribution are compared under different distributions.Simulate sample data generated by normal distribution,skew normal distribution and Chi-square distribution.Take four different sample sizes as N=20,50,100 and200 to calculate the deviation and mean square error of the statistics of the four distributions.Then,the maximum likelihood estimates of the normal distribution,skew normal distribution,t distribution and skew-t distribution are compared based on which distribution is closer to the sample value.It is noted that maximum likelihood estimation is a kind of large sample estimation.When the selected sample size is large enough to a certain extent,maximum likelihood estimation is the best.With the increase of sample size,maximum likelihood estimation will get better and better.Based on the hypothetical sample size and simulated data in this thesis,the maximum likelihood estimators of the four distribution statistics were compared after 500 independent repeated experiments,and it was found that the maximum likelihood estimator based on the normal distribution was closer to the sample value.This shows that it is not sensitive to the original distribution type of the simulated data when only the sample mean and variance are considered.At the same time,the fitting effects of normal distribution,skew normal distribution,t distribution and skew-t distribution on the simulation data generated by normal distribution,skew normal distribution and Chi-square distribution are compared and analyzed.According to the BIC value,it is found that the fitted distributions are normal distribution,skew normal distribution and skew-t distribution respectively.The results show the advantage of judging the original distribution type according to the BIC criterionSecondly,by means of simulated data,we compare the difference between normal distribution,skew-t distribution and skew normal distribution in selecting the number of factors in factor analysis.According to skew normal distribution,skew-t distribution and normal distribution of the three distribution generated by simulation data set,with the scree plot and Kaiser criterion to the number of factors,the distribution of three kinds of choices are found skew normal distribution,skew-t distribution and normal distribution in the choice has no obvious difference between the factors and use these three distribution for factor analysis the effect of no significant difference.Finally,the economic loss caused by catastrophe risk in China is analyzed by using the skew normal distribution and skew-t distribution,and the superiority of the distribution is investigated by using BIC criterion and correlation coefficientR~2.Comparative analysis shows that the fitting effect of normal distribution and skew normal distribution is not ideal,and skew-t distribution is better for fitting effect of catastrophe loss in China. |
