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 the empirical distribution function based on the sample.
Let be an estimator for . Then is an estimator for .
If there exists a such that , then is called the minimum-distance estimate of .
(Drossos & Philippou 1980, p. 121)
Statistics used in estimation
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.
Cramér–von Mises 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).
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 Cramér–von Mises criterion, the Kolmogorov–Smirnov test and the Anderson–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.
- Boos, Dennis D. (1982). "Minimum anderson-darling estimation". Communications in Statistics – Theory and Methods. Taylor & Francis. 11 (24): 2747–2774. doi:10.1080/03610928208828420. ISSN 0361-0926.
- Blyth, Colin R. (June 1970). "On the Inference and Decision Models of Statistics" (PDF). The Annals of Mathematical Statistics. Institute of Mathematical Statistics. 41 (3): 1034–1058. doi:10.1214/aoms/1177696980. ISSN 0020-3157. Retrieved 2008-09-24.
- Drossos, Constantine A.; Philippou, Andreas N. (December 1980). "A Note on Minimum Distance Estimates" (PDF). Annals of the Institute of Statistical Mathematics. Institute of Statistical Mathematics. 32 (1): 121–123. doi:10.1007/BF02480318. ISSN 0020-3157. Retrieved February 18, 2013.CS1 maint: ref=harv (link)
- Parr, William C.; Schucany, William R. (1980). "Minimum Distance and Robust Estimation". Journal of the American Statistical Association. American Statistical Association. 75 (371): 616–624. doi:10.1080/01621459.1980.10477522. ISSN 0162-1459. JSTOR 2287658.CS1 maint: ref=harv (link)
- Wolfowitz, J. (March 1957). "The minimum distance method" (PDF). The Annals of Mathematical Statistics. Institute of Mathematical Statistics. 28 (1): 75–88. doi:10.1214/aoms/1177707038. ISSN 0020-3157. Retrieved February 18, 2013.