Abstract
This review is about verifying and generalizing the supremum test statistic developed by Balakrishnan et al. Exhaustive simulation studies are conducted for various dimensions to determine the effect, in terms of empirical size, of the supremum test statistic developed by Balakrishnan et al. to test multivariate skew-normality. Monte Carlo simulation studies indicate that the Type-I error of the supremum test can be controlled reasonably well for various dimensions for given nominal significance levels 0.05 and 0.01. Cut-off values are provided for the number of samples required to attain the nominal significance levels 0.05 and 0.01. Some new and relevant information of the supremum test statistic are reported here.
Highlights
In this brief review, we summarize existing tests for multivariate skew-normality, both comparing their efficacy in terms of power as well as their practical implementation in terms of programming complexity
Afterwards, we focus on the superb supremum test developed by Balakrishnan et al [1]
We see from this table a sample size between 100 and 200 is required for the supremum test statistic to achieve the nominal significance level (NSL) when d = 3, and the critical value (CV) is obtained via the approximate distribution function, it being apparent that the NSL is always attained when the critical value is obtained via simulation
Summary
We summarize existing tests for multivariate skew-normality, both comparing their efficacy in terms of power as well as their practical implementation in terms of programming complexity. When the assumption of multivariate normality is met, a multivariate linear model with an unstructured variance–covariance matrix can be fit effortlessly due to the existence of closed-form maximum likelihood estimators (MLEs) for the unknown parameters Due to this convenience, some leniency crept into the use of the multivariate normal distribution in order to avoid complexities arising in optimization procedures for multivariate non-normal distributions. FZ(x) = 2φd(x; Ω )Φ(a x), x ∈ Rd, where Ωis a positive-definite (PD) correlation matrix; a is a d-dimensional vector of d×d shape (skewness) parameters, albeit indirectly; φd(Z; Σ) denotes the p.d.f. of a Nd(0d, Σ) random variable; and Φ(·) is the cumulative distribution function of a standard normal random variable This family of distributions can be extended to include location and scale parameters through the usual transformation. An affine transformation of Y, Y∗, is said to be in canonical form if Y∗ ∼ SNd(0d, Id, aY∗ )
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