多维偏正态分布拟合真实数据的效果研究

Research on the Real Data Fitting Effect of Multi-dimensional Skew Normal Distribution

在实际应用中,正态分布普遍应用于数学、物理学等诸多领域,而经济学、医学、社会科学等领域的研究数据往往都不具备严格的对称性,不适合用正态分布来拟合此类非对称数据,因此,各种偏态分布被相继提出。基于上述背景,本文收集了来自人口、经济、金融、医学等多领域的119组多维真实数据,分别利用最大似然估计与改进的EM算法,对多维正态分布与多维偏正态分布进行了参数估计,在此基础上分别从数据的期望、协方差、M型多维偏度与峰度、分布拟合的AIC和BIC值等多方面入手,比较了多维正态分布与多维偏正态分布对真实数据的拟合情况。结果表明,首先,两种分布下期望、协方差的估计值虽有差异,但其相对误差在一定范围内浮动;其次,偏正态分布假设下M型多维偏度及峰度数值较高;同时,偏正态分布将产生更小AIC、BIC值,因此可认为偏正态分布更适合拟合多维真实数据。此外,本文还研究了稳健转换对分布拟合的影响,得出“利用稳健转换后的数据对分布进行拟合通常会得到更小的AIC和BIC值”的结论。

Normal distribution is widely used in mathematics and physics,but the research data in economics,medicine and social sciences are not strictly symmetric in most cases.Therefore,normal distribution is not suitable to fit such asymmetric data.In this study,119 groups of multi-dimensional real data from multiple fields including population,economy,finance and medicine are collected.The maximum likelihood estimation and the improved EM algorithm are used to estimate the unknown parameters of the multi-dimensional normal and skew-normal distribution.On this basis,the fitting of multi-dimensional normal distribution and multi-dimensional skew normal distribution to real data is compared from data expectation,covariance,and Mardia's measurements of skewness and kurtosis,the AIC and BIC values of distribution fitting.The results show that:firstly,although the estimated values of expectation and covariance under the two distributions are different,their relative errors fluctuate within a certain range;secondly,the Mardia's measurements of skewness and kurtosis values are higher under the assumption of the skew normal distribution;finally,the skew normal distribution will produce smaller AIC and BIC values,so skew normal distribution is more suitable for fitting multidimensional real data.In addition,this study also investigates the effect of the robust transformation on distribution fitting.The results show that using the data after robust transformation to fit the two distributions will usually get smaller AIC and BIC values.