Summary
In this paper, we obtain a strong law and central limit theorem for the median deviation under only very mild smoothness conditions on the underlying distribution. Under an additional condition implied by symmetry, we derive a weak Bahadur representation for the median deviation and establish the asymptotic equivalence of the median deviation and the semi-interquartile range.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ghosh, J. K. (1971). A new proof of the Bahadur representation of quantiles and an application,Ann. Math. Statist.,42, 1957–1961.
Hampel, F. R. (1968).Contributions to the Theory of Robust Estimation, Ph.D. dissertation, University of California.
Hampel, F. R. (1974). The influence curve and its role in robust estimation,J. Amer. Statist. Ass.,69, 383–397.
Huber, P. J. (1967). The behaviour of maximum likelihood estimates under non-standard conditions, inProc. Fifth Berkeley Symp. Math. Statist. Prob.,1, 221–233, University of California Press.
Huber, P. J. (1981).Robust Statistics, Wiley, New York.
Rivest, L-P. (1982). Some asymptotic distributions in the location-scale model,Ann. Inst. Statist. Math.,34, 225–239.
Serfling, R. J. (1980).Approximation Theorems of Mathematical Statistics, Wiley, New York.
Author information
Authors and Affiliations
About this article
Cite this article
Hall, P., Welsh, A.H. Limit theorems for the median deviation. Ann Inst Stat Math 37, 27–36 (1985). https://doi.org/10.1007/BF02481078
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02481078