Minimum distance estimation

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Minimum distance estimation (MDE) is a statistical method for fitting a mathematical model to data, usually the empirical distribution.

Contents 1 Definition 2 Goodness of fit statistics based on minimum distance estimation 2.1 Chi-square test 2.2 Cramér-von-Mises criterion 2.3 Kolmogorov-Smirnov test 2.4 Anderson-Darling test 3 See also 4 References

[edit] Definition

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 , . is called the minimum distance estimate of .

[edit] Goodness of fit statistics based on minimum distance estimation

The theory of minimum distance estimation underlies various statistical goodness of fit tests. The difference between the various tests is usually the selected "distance" estimator, or criterion. Below are some examples of statistical tests that rely on minimum distance estimation.

[edit] Chi-square test

The chi-square test uses as its criterion the squared difference between the empirical distribution and the estimate, weighted by the estimate for that group.

[edit] Cramér-von-Mises criterion

The Cramér-von-Mises criterion uses the squared difference between the empirical distribution and the estimate.

[edit] Kolmogorov-Smirnov test

The Kolmogorov-Smirnov test uses the supremum of the absolute difference between the empirical distribution and the estimate.

[edit] Anderson-Darling test

The Anderson-Darling test is similar to Kolmogorov-Smirnov, but instead of using the point of maximal difference between the empirical distribution and the estimate, it uses a smooth weighting based on integrating the difference over the entire interval and then weighting by the reciprocal of the variance.

[edit] See also

• Maximum likelihood estimation

[edit] References

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