Let be an independent and identically distributed (iid) random sample from a population with distribution and .
Let be the empirical distribution function based on the sample.
Let be an estimator for . Then is an estimator for .
Let be a functional returning some measure of "distance" between the two arguments. The functional is also called the criterion function.
If there exists a such that , then is called the minimum-distance estimate of .
(Drossos & Philippou 1980, p. 121)
Most theoretical studies of minimum-distance estimation, and most applications, make use of "distance" measures which underlie already-established goodness of fit tests: the test statistic used in one of these tests is used as the distance measure to be minimised. Below are some examples of statistical tests that have been used for minimum-distance estimation.
Chi-square criterion
The chi-square test uses as its criterion the sum, over predefined groups, of the squared difference between the increases of the empirical distribution and the estimated distribution, weighted by the increase in the estimate for that group.
Anderson–Darling criterion
The Anderson–Darling test is similar to the Cramér–von Mises criterion except that the integral is of a weighted version of the squared difference, where the weighting relates the variance of the empirical distribution function (Parr & Schucany 1980, p. 616).