One-sample
Methoda   MVT/LME   AUC   L2D   EXC (XUV and XMV)
H 0  α1 = … = αm = 0  α1 + … + αm = 0  ( α  1  2   + … + α  m  2    )   1 ∕ 2   = 0  α1 = … = αm
Dimensions in ℝm  0  m−1  m−1  1
DFs for F-statisticb  m, n−m−q+1  1, n−q  1, n−q  m−1, (m−1)(n−q)
Geometric representationc of H0 and H1 (m = 2)
Geometric representationd of HDR when detection failure occurs due to improper H0 formulation  no   no
Two-sample or paired
Method   MVT   AUC   L2D   EXC (XUV and XMV)
H 0  α11 = α21, …, α1m = α2m  ∑  j = 1  m   α  1 j   = ∑  j = 1  m   α  2 j  ( ∑  j = 1  m   α  1 j  2    )   1 ∕ 2   = ( ∑  j = 1  m   α  2 j  2    )   1 ∕ 2  α11 − α21 = … = α1m − α2m
Dimensions in ℝm  0  m−1  m−1  1
DFs for F-statistic  m, n−m−q+1  1, n−q  1, n−q  m−1, (m−1)(n−q)
Geometric representatione of H0 and H1
Geometric representationf of HDR when detection failure occurs due to improper H0 formulation  no