  Many versions of cross-validation (CV) exist in the literature; and each
version though has different variants. All are used interchangeably by many
practitioners; yet, without explanation to the connection or difference among
them. This article has three contributions. First, it starts by mathematical
formalization of these different versions and variants that estimate the error
rate and the Area Under the ROC Curve (AUC) of a classification rule, to show
the connection and difference among them. Second, we prove some of their
properties and prove that many variants are either redundant or "not smooth".
Hence, we suggest to abandon all redundant versions and variants and only keep
the leave-one-out, the $K$-fold, and the repeated $K$-fold. We show that the
latter is the only among the three versions that is "smooth" and hence looks
mathematically like estimating the mean performance of the classification
rules. However, empirically, for the known phenomenon of "weak correlation",
which we explain mathematically and experimentally, it estimates both
conditional and mean performance almost with the same accuracy. Third, we
conclude the article with suggesting two research points that may answer the
remaining question of whether we can come up with a finalist among the three
estimators: (1) a comparative study, that is much more comprehensive than those
available in literature and conclude no overall winner, is needed to consider a
wide range of distributions, datasets, and classifiers including complex ones
obtained via the recent deep learning approach. (2) we sketch the path of
deriving a rigorous method for estimating the variance of the only "smooth"
version, repeated $K$-fold CV, rather than those ad-hoc methods available in
the literature that ignore the covariance structure among the folds of CV.
