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Min-max asymptotic variance of M-estimates of location when scale is unknown

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  • Li, Bing
  • Zamar, Ruben H.

Abstract

This paper extends Huber's (1964) min-max result to the case when the scale parameter is unknown and must be estimated along with the location parameter. A min-max problem in which nature chooses F from a family of symmetric distribution functions around a given location-scale central model, the statistician chooses an M-estimate of location, that is, specifies the influence curve or score function [psi] and the auxiliary scale estimate sn, is solved. The optimal choise for sn is an M-estimate of scale applied to the residuals about the median. The optimal choice for the score function [psi] is a truncated and rescaled maximum likelihood score function for the central model. In he Gaussian case rescaling is not necessary and so, except for the truncation point which is now smaller, Huber's (1964) classical result obtains.

Suggested Citation

  • Li, Bing & Zamar, Ruben H., 1991. "Min-max asymptotic variance of M-estimates of location when scale is unknown," Statistics & Probability Letters, Elsevier, vol. 11(2), pages 139-145, February.
    • Handle: RePEc:eee:stapro:v:11:y:1991:i:2:p:139-145
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    Cited by:

    1. Despina Dasiou & Chronis Moyssiadis, 2001. "The 50% breakdown point in simultaneous M-estimation of location and scale," Statistical Papers, Springer, vol. 42(2), pages 243-252, April.
    2. Kenneth Rice & David Spiegelhalter, 2006. "A Simple Diagnostic Plot Connecting Robust Estimation, Outlier Detection, and False Discovery Rates," Journal of Applied Statistics, Taylor & Francis Journals, vol. 33(10), pages 1131-1147.

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