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### M-estimates for location and scale

Huber (1964) introduced the class of M-estimates. Suppose that are independent and identically distributed with density . The M-estimate of is

where is a user-chosen function. When , Tn is the maximum likelihood estimate of . We can also define Tn implicitly by the equation

where . Assume the following location model. Let be a random sample satisfying

 (2)

where are independent errors with common distribution F0 and finite variance . The parameter of interest is . The cumulative distribution function F of the xi's satisfies . In this case it is natural to take to be a function of a single argument, namely the difference . The defining equation becomes

 (3)

One important drawback of this definition is that in general, if b denotes a real number,

That is, the resulting estimate might not be scale equivariant. To overcome this problem we couple our equation with a scale estimate that satisfies

and instead of using equation (3) we solve

 (4)

Possible choices for Sn are the M-estimates of scale

 (5)

where b is a constant and is again a user-chosen function. The procedure of simultaneously solving equations (4) and (5) is known in the literature as Huber's Proposal 2 (see Huber, 1964). However, on top of the numerical inconvenience of solving this system of two non-linear equations, the robustness properties of the resulting estimates are not satisfactory. Another choice for the scale estimator is to solve the following equation independently of (4)

 (6)

where is an auxiliary location estimate that is generally different from Tn in (4). For example, we can use . Under very mild conditions the estimates Tn and Sn are strongly consistent, i.e., and almost surely. It is well known (see for example Huber, 1981) that if is odd, is even and the underlying distribution is symmetric, Sn and Tn are asymptotically independent and Tn satisfies

with asymptotic variance given by

 (7)

Next: GM-estimates for Regression Up: Robust Point Estimation Previous: Robust Point Estimation
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2000-05-29