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 Statistics used in estimation
3 Theoretical results
4 See also
5 References

Definition

Let $\displaystyle X_1,\ldots,X_n$ be an independent and identically distributed (iid) random sample from a population with distribution $F(x;\theta)\colon \theta\in\Theta$ and $\Theta\subseteq\mathbb{R}^k (k\geq 1)$ .

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

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

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

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

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.

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.

Cramér–von Mises criterion

The Cramér–von Mises criterion uses the integral of the squared difference between the empirical and the estimated distribution functions.

Kolmogorov–Smirnov criterion

The Kolmogorov–Smirnov test uses the supremum of the absolute difference between the empirical and the estimated distribution functions.

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.

Theoretical results

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.

References

Boos, D.D. (1982). "Minimum Anderson–Darling Estimation". Communications in Statistics, Part A – Theory and Methods, 11 (24), 2747–2774.
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. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoms/1177696980. Retrieved 2008-09-24.
Drossos, Constantine A.; Andreas N. Philippou (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. http://www.ism.ac.jp/editsec/aism/pdf/032_1_0121.pdf. Retrieved 2008-09-24.

Parr W.C., Schucany W.R. (1980). "Minimum distance and robust estimation". Journal of the American Statistical Association, 75, 616–624.

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. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoms/1177707038. Retrieved 2008-09-24.

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) · Median · Mode

Dispersion	Range · Standard deviation · Coefficient of variation · Percentile · Interquartile range

Shape	Variance · Skewness · Kurtosis · Moments · L-moments

Count data

Index of dispersion

Summary tables

Grouped data · Frequency distribution · Contingency table

Dependence

Pearson product-moment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Partial correlation · Scatter plot

Statistical graphics

Bar chart · Biplot · Box plot · Control chart · Correlogram · Forest plot · Histogram · Q-Q plot · Run chart · Scatter plot · Stemplot · Radar chart

Data collection

Designing studies	Effect size · Standard error · Statistical power · Sample size determination

Survey methodology	Sampling · Stratified sampling · Opinion poll · Questionnaire

Controlled experiment	Design of experiments · Factorial experiment · Randomized experiment · Random assignment · Replication · Blocking · Optimal design

Uncontrolled studies	Natural experiment · Quasi-experiment · Observational study

Statistical inference

Statistical theory	Sampling distribution · Sufficient statistic · Meta-analysis

Bayesian inference	Bayesian probability · Prior · Posterior · Credible interval · Bayes factor · Bayesian estimator · Maximum posterior estimator

Frequentist inference	Confidence interval · Hypothesis testing · Likelihood-ratio

Specific tests	Z-test (normal) · Student's t-test · F-test · Pearson's chi-squared test · Wald test · Mann–Whitney U · Shapiro–Wilk · Signed-rank · Kolmogorov–Smirnov test

General estimation	Mean-unbiased · Median-unbiased · Maximum likelihood · Method of moments · Minimum distance · Density estimation

Correlation and regression analysis

Categorical, multivariate, time-series, or survival analysis

Applications

Category · Portal · Outline · Index

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