Minimum-distance estimation (MDE) is a conceptual method for fitting a statistical model to data, usually the empirical distribution. Often-used estimators such as ordinary least squares can be thought of as special cases of minimum-distance estimation.
While consistent and asymptotically normal, minimum-distance estimators are generally not statistically efficient when compared to maximum likelihood estimators, because they omit the Jacobian usually present in the likelihood function. This, however, substantially reduces the computational complexity of the optimization problem.
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 its two arguments. The functional is also called the criterion function.
If there exists a such that , then is called the minimum-distance estimate of .
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.
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.
The CramérâÂÂvon Mises criterion uses the integral of the squared difference between the empirical and the estimated distribution functions .
The KolmogorovâÂÂSmirnov test uses the supremum of the absolute difference between the empirical and the estimated distribution functions .
The AndersonâÂÂDarling test is similar to the CramérâÂÂvon 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 .
The theory of minimum-distance estimation is related to that for the asymptotic distribution of the corresponding statistical goodness of fit tests. Often the cases of the CramérâÂÂvon Mises criterion, the KolmogorovâÂÂSmirnov test and the AndersonâÂÂDarling test are treated simultaneously by treating them as special cases of a more general formulation of a distance measure. Examples of the theoretical results that are available are: consistency of the parameter estimates; the asymptotic covariance matrices of the parameter estimates.