# mvnTest: A Powerful Test for Multivariate Normality (Zhou-Shao's Test) In mvnormalTest: Powerful Tests for Multivariate Normality

## Description

A simple and powerful test for multivariate normality with a combination of multivariate kurtosis (MK) and Shapiro-Wilk which was proposed by Zhou and Shao (2014). The p-value of the test statistic (T_n) is computed based on a simulated null distribution of T_n. Details see Zhou and Shao (2014).

## Usage

 `1` ```mvnTest(X, B = 1000, pct = c(0.01, 0.99)) ```

## Arguments

 `X` an n*p data matrix or data frame, where n is number of rows (observations) and p is number of columns (variables) and n>p. `B` number of Monte Carlo simulations for null distribution, default is 1000 (increase B to increase the precision of p-value). `pct` percentiles of MK to get c_1 and c_2 described in the reference paper, default is (0.01, 0.99).

## Value

Returns a list with two objects:

`mv.test`

results of the Zhou-Shao's test for multivariate normality , i.e., test statistic T_n, p-value (under H0, i.e. multivariate normal, that T_n is at least as extreme as the observed value), and multivariate normality summary (YES, if p-value>0.05).

`uv.shapiro`

a dataframe with p rows detailing univariate Shapiro-Wilk tests. Columns in the dataframe contain test statistics W, p-value,and univariate normality summary (YES, if p-value>0.05).

## References

Zhou, M., & Shao, Y. (2014). A powerful test for multivariate normality. Journal of applied statistics, 41(2), 351-363.

Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611.

`power.mvnTest`, `msk`, `mardia`, `msw`, `faTest`, `mhz`
 ``` 1 2 3 4 5 6 7 8 9 10``` ```set.seed(12345) ## Data from gamma distribution ## X = matrix(rgamma(50*4,shape = 2),50) mvnTest(X, B=100) ## load the ubiquitous multivariate iris data ## ## (first 50 observations of columns 1:4) ## iris.df = iris[1:50, 1:4] mvnTest(iris.df, B=100) ```