Optimal robust estimates using the Hellinger distance. (English) Zbl 1284.62202

Marazzi, Alfio; Yohai, Victor J.

Adv. Data Anal. Classif., ADAC 4, No. 2-3, 169-179 (2010).

Summary: Optimal robust M-estimates of a multidimensional parameter are described using Hampel’s infinitesimal approach. The optimal estimates are derived by minimizing a measure of efficiency under the model, subject to a bounded measure of infinitesimal robustness. To this purpose we define measures of efficiency and infinitesimal sensitivity based on the Hellinger distance. We show that these two measures coincide with similar ones defined by Yohai using the Kullback-Leibler divergence, and therefore the corresponding optimal estimates coincide too. We also give an example where we fit a negative binomial distribution to a real dataset of “days of stay in hospital” using the optimal robust estimates.

MSC:

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

Keywords:

Hampel’s infinitesimal approach; gross error sensitivity; negative binomial distribution

Software:

robustbase

References:

[1]	Beran R (1977a) Robust location estimates. Ann Stat 5: 431–444 · Zbl 0381.62032
[2]	Beran R (1977b) Minimum Hellinger distance estimates for parametric models. Ann Stat 5: 445–463 · Zbl 0381.62028
[3]	Beran R (1978) An efficient and robust adaptive estimator of location. Ann Stat 6: 292–313 · Zbl 0378.62051
[4]	Gervini D, Yohai VJ (2002) A class of robust and fully efficient regression estimates. Ann Stat 30(2): 583–616 · Zbl 1012.62073
[5]	Hampel FR (1974) The influence curve and its role in robust estimation. J Am Stat Assoc 69: 383–394 · Zbl 0305.62031
[6]	Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics: the approach based on influence functions. Wiley, New York
[7]	Hilbe JM (2008) Negative binomial regression. Cambridge University Press, Cambridge
[8]	Marazzi A, Yohai VJ (2004) Adaptively truncated maximum likelihood regression with asymmetric errors. J Stat Plan Inference 122: 271–291 · Zbl 1040.62057
[9]	Maronna RA, Martin RD, Yohai VJ (2006) Robust statistics: theory and methods. Wiley, Chichister
[10]	Stahel WA (1981) Robust estimation, infinitesimal optimality and covariance matrix estimators. Ph.D. Thesis, ETH, Zurich
[11]	Tamura RN, Boos DD (1986) Minimum Hellinger distance estimation for multivariate location and covariance. J Am Stat Assoc 81: 223–229 · Zbl 0601.62051
[12]	Yohai VJ (2008) Optimal robust estimates using the Kullback–Leibler divergence. Stat Probab Lett 78: 1811–1816 · Zbl 1147.62025