Tests For Normality Based On Skewness And Kurtosis

The test of goodness-of-fit of distributions in the statistical theory occupies an important position,and the normal distribution is the most common statistical distribution. Many statistical theories and models are established under the assumption of random variables (or vectors) subjecting to a one-dimensional (or multi-dimensional) normal distribution. Therefore, the research of the one-dimensional and multi-dimensional goodness-of-fit test for normality is important.This article discusses a class of the goodness-of-fit tests of one-dimensional and multi-dimensional normal distribution. Characteristics of these tests is the use of the skewness and kurtosis test statistics which characterize morphological features of the distribution. Through theoretical analysis of these test methods with simulation and comparison of test results , evaluation and recommendations are given.Normality tests is discussed in one-dimensional and multi-dimensional case respectively. Among them, the construction of one-dimensional normality test statistics are based on one-dimensional sample skewness and kurtosis; For the multidimensional case, according to Mardia, Theilen and Srivastava's three different definitions of multi-dimensional skewness and kurtosis,different multi-dimensional normality test statistics are constructed. This article discusses the nature , digital features and theoretical distribution of the test statistic. Meanwhile,numerical simulations of critical values are also given in tabular form, thus expanding the scope of application of these methods. In addition,this paper has done a comparison research of the results of one-dimensional and multi-dimensional normal tests using Monte Carlo Methods. Conclusions have some value for reference.At last ,this paper presents a new test for multidimensional normality -that is principal component method. This is an innovative design. The main idea of this method is to convert multidimensional normality test into one-dimensional normality test by extracting the principal components. Simulation results show that the test results are good, so it can be applied in statistical analysis and practice.