Minimum-distance estimation (MDE) is a conceptual method for fitting a statistical model to data, usually theempirical distribution. Often-used estimators such asordinary least squares can be thought of asspecial cases of minimum-distance estimation.

Whileconsistent andasymptotically normal, minimum-distance estimators are generally notstatistically efficient when compared tomaximum likelihood estimators, because they omit theJacobian usually present in thelikelihood function. This, however, substantially reduces thecomputational complexity of the optimization problem.

Let${\displaystyle {\displaystyle X_1,\dots ,X_n}}$ be anindependent and identically distributed (iid)randomsample from apopulation withdistribution${\displaystyle F(x;\theta ):\theta \in \mathrm{\Theta }}$ and${\displaystyle \mathrm{\Theta }\subseteq {\mathbb{R}}^k(k\ge 1)}$.

Let${\displaystyle {\displaystyle F_n(x)}}$ be theempirical distribution function based on the sample.

Let${\displaystyle \hat{\theta }}$ be anestimator for${\displaystyle {\displaystyle \theta }}$. Then${\displaystyle F(x;\hat{\theta })}$ is an estimator for${\displaystyle {\displaystyle F(x;\theta )}}$.

Let${\displaystyle d[\cdot ,\cdot ]}$ be afunctional returning some measure of"distance" between its two arguments. The functional${\displaystyle {\displaystyle d}}$ is also called the criterion function.

If there exists a${\displaystyle \hat{\theta }\in \mathrm{\Theta }}$ such that${\displaystyle d[F(x;\hat{\theta }),F_n(x)]=inf\{d[F(x;\theta ),F_n(x)];\theta \in \mathrm{\Theta }\}}$, then${\displaystyle \hat{\theta }}$ is called theminimum-distance estimate of${\displaystyle {\displaystyle \theta }}$.

(Drossos & Philippou 1980, p. 121)

Most theoretical studies of minimum-distance estimation, and most applications, make use of "distance" measures which underlie already-establishedgoodness 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.

Thechi-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.

TheCramér–von Mises criterion uses the integral of the squared difference between the empirical and the estimated distribution functions (Parr & Schucany 1980, p. 616).

TheKolmogorov–Smirnov test uses thesupremum of theabsolute difference between the empirical and the estimated distribution functions (Parr & Schucany 1980, p. 616).

TheAnderson–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).

The theory of minimum-distance estimation is related to that for the asymptotic distribution of the corresponding statisticalgoodness of fit tests. Often the cases of theCramér–von Mises criterion, theKolmogorov–Smirnov test and theAnderson–Darling 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.

• Maximum likelihood estimation
• Maximum spacing estimation

• Estimation methods
• Statistical distance
• Mathematical modeling

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