dvalue

Dvalue is a metric used in statistics and data analysis to quantify the dissimilarity between two distributions, whether empirical or theoretical. It is defined as the average distance between the quantile functions of the two distributions, with the result typically normalized to produce a dimensionless value in a fixed range. In many formulations, for distributions with cumulative distribution functions F and G and with a common range [a, b], Dvalue can be expressed as Dvalue(F, G) = (1/(b − a)) ∫_0^1 |F^{-1}(p) − G^{-1}(p)| dp. The measure is nonnegative, symmetric, and equals zero when the distributions are identical (aside from negligible differences).

Computation of Dvalue often relies on empirical data. When working with samples, the quantile functions are approximated by order statistics or by interpolation of empirical quantiles. The integral is then approximated by numerical integration over a set of evenly spaced p-values. An alternative practical approach uses histograms or kernel density estimates to compute a discrete version of the distance, similar to the discrete version of the Wasserstein distance.

Variants and extensions of Dvalue may include weighting schemes that emphasize specific parts of the distribution, robustness adjustments to handle outliers, or different normalization schemes to accommodate diverse data ranges. In practice, Dvalue is used to compare models, detect distributional shifts over time, assess goodness of fit in a distributional sense, and support domain adaptation tasks where matching target and source distributions is important.

Dvalue is closely related to distributional distance concepts such as the Wasserstein distance and the Kolmogorov–Smirnov statistic. See also Wasserstein distance and Cramér–von Mises criterion for related approaches to comparing distributions.