Smooth Test And Application Of Multivariate Skew-Normality

The multivariate multilinear model has been widely applied in many fields,and it is usually assumed that the errors in the model obey the multivariate normal distribution.Actually,the data has some level of skewness,the multivariate normality assumption is usually violated.The family of multivariate skew-normal(SN)distributions extends the widely employed family of multivariate normal distributions by introducing a vector parameter to capture the skewness in the data.The skew-normal(SN)distributions family preserves many statistical properties of classical multivariate normal distribution family.Firstly,the paper introduces the definition and properties of the multivariate skewnormal distribution,then gives the moment estimation form of the parameters of the multivariate skew-normal distribution provides the statistical simulation experiment results.Based on the smooth test for uniformity on the surface of a unit sphere,the smooth test of the multivariate skew-normality is proposed.Specifically,based on the canonical form of the multivariate skew-normal distribution,the innovation distribution is converted into a spherically symmetric distribution by data conversion,then transformer it into a spherical uniform distribution,and then convert the goodness of fit test of the multivariate skew-normal distribution into a spherical uniform.The bootstrap method is use to estimate the p-value of the test statistic.And the Anderson-Darling goodness-of-fit test method is used to test whether the shape parameter in the multivariate skew normal distribution is zero.In the process of empirical analysis,the first-order difference and logarithmic operation is performed on the data of IBM,Coca-Cola and S&P500 three stocks to obtain the logarithmic return rate of the three stocks.After the ARCH effect test,the GARCH(1,1)model was fitted to the three stocks,and the DCC model was established for the standardized residual sequence of the model.Under the assumption that the innovation distribution of the DCC model obeys the multivariate partial normal distribution,a goodness-of-fit test of the distribution is performed to obtain the parameter estimates of the DCC model.Calculating the time-varying correlation time series diagram between the three asset yields,it can be seen that the correlation between the IBM stock and the SP500 index has been significantly reduced after the 2008 financial crisis.