M-estimates are measures of location that are not as sensitive as the mean to outlier values. With observations x1, x2,…, xn, the sample mean can be characterized as the value of θ that minimizes , where g(u)=u2. The sample median can be characterized in a similar way, though now g(u)=|u|.
M-estimates can be characterized in this same way, but the functional forms for g are chosen to be less sensitive to outlier values. One frequently used alternative as a measure of location is the Huber function: where k is a tuning constant (often set equal to twice the median absolute deviation).
A second alternative is the biweight function: where k is again a tuning constant and is here often set equal to seven times the median absolute deviation. See also L-estimate.
Subjects: Probability and Statistics.