Smooth Test For Error Distribution In Multivariate Linear Model

The test of mult ivariate normal distribution data has been a hot research topic in the field of statistics. The common bayes discrimination is discussed under the hypothesis of the normal distribution. Failure of the population distribution means failure of the model, the conclusions according to the model are then invalid. Error distribution assumption is multivariate normal distribution in classic multivariate linear regression model. So the importance of multivariate normality test for error distribution in multivariate linear model is self-evident.The basic hypothesis of classic multivariate analysis is the multivariate normal distribution, thus the goodness-of-fit test for multivariate normal distribution is o f great importance. The elliptically symmetric distribution contains Laplace distribution multivariate Laplace distribution, multivariate t distribution, multivariate normal distribution and other distribution. While the elliptically symmetric distribution can be constructed by uniformity on the surface of a unit sphere. Then, the uniform distribution on the surface of a unit sphere is a foundational distribution in multivariate distribution. So the goodness-of-fit test for multivariate normal distribution can be translated into the change to the goodness-of-fit test for uniform distribution on the surface of a unit sphere. Therefore, goodness-of-fit test for uniformity on the surface of a unit sphere is particularly important.In this paper, we introduce the basic theory of smooth test and its application in multivariate normal distribution. After a certain correction, the smooth test statistics is applied to the multivariate normality test for error distribution in multivariate linear model. Thereby the multivariate linear model can be test by goodness-of-fit test. The main conclusions are as follows:(1) Based on the spherical harmonics. the smooth test for multivariate normal distribution is proposed. The algorithm steps of the smooth test statistics and the simulate critical values are given. The smooth test statistics of multivariate normal distribution has carried on power simulation.(2) The mult ivariate normalit y smooth test for error distribution in mult ivariate linear mode is proposed. The algorithm steps of the smooth test statistics for error distribution and the simulate crit ical values are given. The smooth test statistics for error distribution has carried on power simulation.(3) Using the relationship between GDP structure and consumption of GDP, we set up a multivariate linear regression model. Applying the smooth test statistics for error distribution in(2), the new multivariate linear regression model has carried on normalit y test. Meanwhile, Combining with the actual economic data, we use correction model to get the prediction of out-of-sample and predict ellipsoid.