Minimum distance estimation

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

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 $\displaystyle \theta$ .

(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.

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 (Parr & Schucany 1980, p. 616).

Kolmogorov–Smirnov criterion

The Kolmogorov–Smirnov test uses the supremum of the absolute difference between the empirical and the estimated distribution functions (Parr & Schucany 1980, p. 616).

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 (Parr & Schucany 1980, p. 616).

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, 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.
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.
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.

Statistics

Descriptive statistics

Continuous data

Center	Mean arithmetic geometric harmonic Median Mode

Dispersion	Variance Standard deviation Coefficient of variation Percentile Range Interquartile range

Shape	Moments Skewness Kurtosis L-moments

Count data

Index of dispersion

Summary tables

Dependence

Graphics

Data collection

Study design	Population Statistic Effect size Statistical power Sample size determination Missing data

Survey methodology	Sampling Standard error stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in

Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife

Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F

Goodness of fit	Chi-squared Kolmogorov–Smirnov Anderson–Darling Normality (Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC

Rank statistics	Sign Sample median Signed rank (Wilcoxon) Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova 1-way (Kruskal–Wallis) 2-way (Friedman) Ordered alternative (Jonckheere–Terpstra)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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