Finite sample breakdown of M- and P-estimators (Q761729)

The breakdown point of an estimator is the smallest fraction of bad sample values that may cause the estimator to take on arbitrarily large values. In this paper the author investigates the finite sample breakdown point properties of M-estimators T defined by \(\sum \rho (x_ i- T)=^{!}\min\), and of the associated Pitman-type or P-estimators, defined by \[ T_ P=\int \exp \{-\Sigma \rho (x_ i-\theta \}\theta d\theta)/\int \exp \{-\Sigma \rho (x_ i-\theta)\} d\theta. \] If \(\rho\) is symmetric and \(\psi =\rho '\) is monotone and bounded, then the breakdown point of either estimator is \(\epsilon^*=1/2\). If \(\psi\) decreases to 0 for large x, the same remains true if \(\rho\) is unbounded. For bounded \(\rho\), the P-estimator is undefined, and the breakdown point of the M-estimator is typically slightly less than 1/2; it is calculated in explicit form.