Alternatives to the median absolute deviation.

*(English)*Zbl 0792.62025Summary: 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.

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.