 Scale M-estimator

A Scale M-estimator estimates the spread of centered data by solvingfor a chosen function $\psi$. If the center is estimated, residuals $x_i-\hat{\mu }$ replace $x_i$. The function and its normalization specify the population scale being estimated; an arbitrary choice need not have a positive or unique root.

For known center zero and at least one nonzero observation, choosing $\psi (u)=u^2-1$ gives $t=\sqrt{n^{-1}{\sum }_i{x}_i^2}$, the root mean square. This example is highly sensitive to an outlier. Bounded choices of $\psi$ can instead limit an observation's contribution to the estimating equation, but their influence function and breakdown point still require analysis.

If $\psi$ is an even function and the positive root is unique, multiplying the data by $c\ne 0$ multiplies the root by $|c|$: substitute $t_c=|c|t$ into the equation. This is scale equivariance, the expected response to changing measurement units.