Minimum distance estimation

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
* 2.1 Chi-square criterion
* 2.2 Cramér–von Mises criterion
* 2.3 Kolmogorov–Smirnov criterion
* 2.4 Anderson–Darling criterion
* 3 Theoretical results
* 4 See also
* 5 References

## Definition

Let X1,…,Xn be an independent and identically distributed (iid) random sample from a population with distribution F(x;θ):θ∈Θ and Θ⊆ℝk(k≥1).

Let Fn(x) be the empirical distribution function based on the sample.

Let θ^ be an estimator for θ. Then F(x;θ^) is an estimator for F(x;θ).

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

If there exists a θ^∈Θ such that d[F(x;θ^),Fn(x)]=inf⁡{d[F(x;θ),Fn(x)];θ∈Θ}, then θ^ is called the minimum distance estimate of θ.

(Drossos Philippou)

## 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).

### Kolmogorov–Smirnov criterion

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

### 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).

## 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.

## See also

* Maximum likelihood estimation
* Maximum spacing estimation

## 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. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoms/1177696980. 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. http://www.ism.ac.jp/editsec/aism/pdf/032_1_0121.pdf. 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. http://www.dtic.mil/get-tr-doc/pdf?AD=ADA085183.
* 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 February 18, 2013.

* v
* t
* e

Statistics

* Outline
* Index

Descriptive statistics

| Continuous data|

| Center|

* Mean
* arithmetic
* geometric
* harmonic
* Median
* Mode

|

Dispersion|

* Variance
* Standard deviation
* Coefficient of variation
* Percentile
* Range
* Interquartile range

Shape|

* Central limit theorem
* Moments
* Skewness
* Kurtosis
* 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

Graphics|

* Bar chart
* Biplot
* Box plot
* Control chart
* Correlogram
* Fan chart
* Forest plot
* Histogram
* Pie chart
* Q–Q plot
* Run chart
* Scatter plot
* Stem-and-leaf display
* Radar chart
* Violin plot

Data collection

| Study design|

* Population
* Statistic
* Effect size
* Statistical power
* Optimal design
* Sample size determination
* Replication
* Missing data

|

Survey methodology|

* Sampling
* stratified
* cluster
* Standard error
* Opinion poll
* Questionnaire

Controlled experiments|

* Scientific control
* Randomized experiment
* Randomized controlled trial
* Random assignment
* Blocking
* Interaction
* Factorial experiment

Adaptive Designs|

* Adaptive clinical trial
* Up-and-Down Designs
* Stochastic approximation

Observational Studies|

* Cross-sectional study
* Cohort study
* Natural experiment
* Quasi-experiment

Statistical inference

| Statistical theory|

* Population
* Statistic
* Probability distribution
* Sampling distribution
* Order statistic
* Empirical distribution
* Density estimation
* Statistical model
* Model specification
* Lp space
* Parameter
* location
* scale
* shape
* Parametric family
* Likelihood (monotone)
* Location–scale family
* Exponential family
* Completeness
* Sufficiency
* Statistical functional
* Bootstrap
* U
* V
* Optimal decision
* loss function
* Efficiency
* Statistical distance
* divergence
* Asymptotics
* Robustness

|

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
* Score/Lagrange multiplier
* Wald

Specific tests|

* Z-test (normal)
* Student's t-test
* F-test

Goodness of fit|

* Chi-squared
* G-test
* Kolmogorov–Smirnov
* Anderson–Darling
* Lilliefors
* Jarque–Bera
* 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|

* Bayesian probability
* prior
* posterior
* Credible interval
* Bayes factor
* Bayesian estimator
* Maximum posterior estimator

* Correlation
* Regression analysis

| Correlation|

* Pearson product-moment
* Partial correlation
* Confounding variable
* Coefficient of determination

|

Regression analysis|

* Errors and residuals
* Regression 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|

* Cohen's kappa
* Contingency table
* Graphical model
* Log-linear model
* McNemar's test

|

Multivariate|

* Regression
* Manova
* Principal components
* Canonical correlation
* Discriminant analysis
* Cluster analysis
* Classification
* Structural equation model
* Factor analysis
* Multivariate distributions
* Elliptical distributions
* Normal

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
* Whittle likelihood

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
* Jurimetrics
* National accounts
* Official statistics
* Population statistics
* Psychometrics

Spatial statistics|

* Cartography
* Environmental statistics
* Geographic information system
* Geostatistics
* Kriging

* Category
* '
* Commons
* WikiProject

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