{c}

A <Scale M-estimator> estimates the spread of centered data by solving
$$
\sum_{i=1}^n\psi(x_i/t)=0,\qquad t>0,
$$
for a chosen function $\psi$. If the center is estimated, residuals $x_i-\widehat\mu$ 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 $c\ne0$ multiplies the root by $|c|$: substitute $t_c=|c|t$ into the equation. This is scale <equivariance>, the expected response to changing measurement units.


 Scale M-estimator (source code)