Finite Sample Breakdown of $M$- and $P$-Estimators

Abstract

The finite sample breakdown properties of $M$-estimators, defined by $\sum\rho(x_i - T) = \min!$, and of the associated Pitman-type or $P$-estimators, defined by $T = \frac{\int \exp\{-\Sigma \rho(x_i - \theta)\}\theta d\theta}{\int \exp\{-\Sigma\rho(x_i - \theta)\} d\theta},$ are investigated. If $\rho$ is symmetric, and $\psi = \rho'$ is monotone and bounded, then the breakdown point of either estimator is $\varepsilon^\ast = 1/2$. If $\psi$ decreases to 0 for large $x$ ("redescending estimators"), the same result remains true if $\rho$ is unbounded. For bounded $\rho$, the $P$-estimator is undefined, and the breakdown point of the $M$-estimator typically is slightly less than $1/2$; it is calculated in explicit form.