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A Cramér-Rao analogue for median-unbiased estimators. (English) Zbl 0743.62022

Summary: Adopting a measure of dispersion proposed by J. Bejar Alamo [ibid. 15, 93-102 (1964)] and extending the analysis of the second author [Optimum estimation under generalized unbiasedness. Ph. D. diss., Iowa State Univ., Ames (1977)] and of the second and third authors [A robust Cramér-Rao analogue (unpublished)], an analogue of the classical Cramér-Rao lower bound for median-unbiased estimators is developed for absolutely continuous distributions with a single parameter, in which mean-unbiasedness, the Fisher information, and the variance are replaced by median-unbiasedness, the first absolute moment of the sample score, and the reciprocal of twice the median-unbiased estimator’s density height evaluated at its median point.
We exhibit location-parameter and scale-parameter families for which there exist median-unbiased estimators meeting the bound. We also give an analogue of the Chapman-Robbins inequality which is free from regularity conditions.

MSC:

62F10 Point estimation
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References:

[1] BEJAR ALAMO, J. (1964): “Estimación en Mediana{”,Trabajos de Estadística, 15, 93–102.}
[2] CHAPMAN, D. G., and ROBBINS, H. (1951): “Minimum variance estimation without regularity assumptions{”,Ann. Math. Statist., 22, 581–586.} · Zbl 0044.34302 · doi:10.1214/aoms/1177729548
[3] CRAMER, R. (1946):Mathematical methods of statistcs, Princeton University Press, Princeton.
[4] STANGENHAUS, G. (1977):Optimum estimation under generalized unbiasedness, Unpublished Ph. D. dissertation, Iowa State University, Ames, Iowa.
[5] STANGENHAUS, G., and DAVID, H. T. (1978): “Estimação Não Tendenciosa em Mediana{”,Proceedings of the 3rd, SINAPE San Paulo, Brazil, 99–102.}
[6] STANGENHAUS, G., and DAVID, H. T. (1978):A robust Cramér-Rao analogue. Unpublished paper Department of Statistics, Iowa State University Ames, Iowa.
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