BHEP multivariate normality test

Given a sample of n independent and identically distributed (i.i.d) observation vectors of dimension d, x1 , x2 , ..., xn, drawn from an unknown continuous distribution function F(x), the Baringhaus–Henze–Epps–Pulley multivariate normality test tests
  H0: F(x) = F0 (x) against H1 : F(x) ≠ F0(x), where

F0(x) is the multivariate normal distribution function Nd(μ, Σ) with mean μ and covariance matrix Σ whose derivative with respect to x is the multivariate probability density function f0(x)= (2π)-d/2 |Σ|½ exp{-½(x-μ)T Σ-1 (x-μ)}.

According to Henze & Wagner (1997) the test statistic in BHEP has a number of favourable characteristics: . it is affine invariant . it is consistent

## Settings

The test depends on a smoothing parameter h that can be chosen in various ways: Henze & Wagner (1997) recommend h = 1.41, while Tenreiro (2009) recommends hs = 0.448 + 0.026 cd for short tailed alternatives and " " hl = 0.928 + 0.049 cd for long tailed alternatives.

### Links to this page

- GaussianMixture & TableOfReal: To Table (BHEP normality tests)...
- TableOfReal: Report multivariate normality (BHEP)...