Robust statistics

Robust statistics studies methods that remain reliable when data contain outliers or depart from assumed statistical models. It measures and limits how strongly unusual observations or small model violations can affect conclusions.

Influence function: A function describing the effect of an infinitesimal change in a data distribution on a statistical estimate. It measures how strongly a single observation can affect an estimator.
Statistical estimator: A rule that uses sample data to estimate an unknown quantity in a statistical model. Robustness describes how an estimator behaves when its data or model assumptions are imperfect.
Classical statistics: Statistical methods developed under specified probability models, often with optimality evaluated when those models hold exactly. Classical procedures can be highly efficient under ideal assumptions but sensitive to small departures.
Robust regression: Regression methods designed to limit the influence of outliers or departures from error-model assumptions. It applies robust estimation to relationships between response and predictor variables.
Breakdown point: The largest fraction of contaminated observations an estimator can tolerate before its value becomes arbitrarily misleading. It quantifies resistance to extreme outliers.
Outlier: An observation that differs substantially from the pattern of the other observations in a dataset. Outliers are a central source of sensitivity that robust methods aim to control.
Least squares: A method that estimates parameters by minimizing the sum of squared residuals. Squaring residuals gives extreme observations disproportionate influence compared with robust alternatives.
Robust covariance estimation: Methods for estimating multivariate location and covariance while limiting sensitivity to contaminated observations. It supports resistant descriptions of multivariate data and outlier detection.
M-estimator: An estimator defined by minimizing a sum of transformed residuals or solving a related estimating equation. M-estimators reduce the influence of large residuals through their chosen loss function.
Contamination model: A probability model representing observed data as mostly drawn from a target distribution with a small fraction from another distribution. It gives a formal account of the data corruption robust methods are designed to withstand.

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