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Alternatives to the median absolute deviation. (English) Zbl 0792.62025
Summary: In robust estimation one frequently needs an initial or auxiliary estimate of scale. For this one usually takes the median absolute deviation $$\text{MAD}_ n= 1.4826 \text{ med}_ i \{| x_ i- \text{med}_ j x_ j|\}$$, because it has a simple explicit formula, needs little computation time, and is very robust as witnessed by its bounded influence function and its 50% breakdown point. But there is still room for improvement in two areas: the fact that $$\text{MAD}_ n$$ is aimed at symmetric distributions and its low (37%) Gaussian efficiency.
We set out to construct explicit and 50% breakdown scale estimators that are more efficient. We consider the estimator $$S_ n= 1.1926 \text{ med}_ i \{\text{med}_ j | x_ i-x_ j|\}$$ and the estimator $$Q_ n$$ given by the .25 quantile of the distances $$\{| x_ i-x_ j|$$; $$i<j\}$$. Note that $$S_ n$$ and $$Q_ n$$ do not need any location estimate. Both $$S_ n$$ and $$Q_ n$$ can be computed using $$O(n\log n)$$ time and $$O(n)$$ storage. The Gaussian efficiency of $$S_ n$$ is 58%, whereas $$Q_ n$$ attains 82%. We study $$S_ n$$ and $$Q_ n$$ by means of their influence functions, their bias curves (for implosion as well as explosion), and their finite-sample performance. Their behavior is also compared at non-Gaussian models, including the negative exponential model where $$S_ n$$ has a lower gross-error sensitivity than the MAD.

##### MSC:
 62F35 Robustness and adaptive procedures (parametric inference) 62F10 Point estimation
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