A New Goodness of Fit Test for Multivariate Normality and Comparative Simulation Study

Abstract

The testing of multivariate normality remains a significant scientific problem. Although it is being extensively researched, it is still unclear how to choose the best test based on the sample size, variance, covariance matrix and others. In order to contribute to this field, a new goodness of fit test for multivariate normality is introduced. This test is based on the mean absolute deviation of the empirical distribution density from the theoretical distribution density. A new test was compared with the most popular tests in terms of empirical power. The power of the tests was estimated for the selected alternative distributions and examined by the Monte Carlo modeling method for the chosen sample sizes and dimensions. Based on the modeling results, it can be concluded that a new test is one of the most powerful tests for checking multivariate normality, especially for smaller samples. In addition, the assumption of normality of two real data sets was checked.

Proposal and Pilot Study: A Generalization of the W or W' Statistic for Multivariate Normality

“…This pilot study only handles one sample size (n = 20) and one number of variables (k = 4) when the variation of n and k would allow the definition of power curves to compare the tests. There are also other multivariate normality tests not covered in the present work, such as those of Cox and Small [55], Henze and Zirkler [56], and Doornik and Hansen [57], available in the R program [58], the Monte Carlo version of multivariate runs test available in Excel [9] [59] or the tests of Arnastauskaite, Ruzgas and Braženas [60] and Kesemen, Tiryaki, Tezel and Özkul [61], more recently. There is also another generalization of the Shapiro-Wilk test developed by Villaseñor-Alva and González-Estrada [62], different from that of Royston [10] and the present work.…”

“…There is also another generalization of the Shapiro-Wilk test developed by Villaseñor-Alva and González-Estrada [62], different from that of Royston [10] and the present work. Any of these tests not included would be excellent comparison options for future research, although there is currently no evidence or consensus on which is the best test [58] [60].…”

Symbolic Regression Approaches for the Direct Calculation of Pipe Diameter

“…The other approaches were successfully tested as well, such as iterative methods [7,8], explicit solutions based on special functions [24,25], and neural networks [37]. Although the goodness of fit is a subject of intensive research [38][39][40][41], the goodness of fit methods are not widely used for flow friction modeling. Anyway, the goodness of fit methods were used to analyze the properties of non-Newtonian fluid samples [42].…”

Model Selection in Generalized Linear Models

“…Model selection has garnered significant attention in the Bayesian approach to generalized linear mixed models [16]. In this context, it is worth noting that there is a large amount of literature on goodness-of-fit tests, for example [17][18][19][20][21][22][23][24]. However, in this paper, we investigate variable selection in GLM for non-symmetric data, such as binomial and Poison regression models, when the dataset is small or moderate.…”