Abstract
Normal 0 false false false EN-US X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Times New Roman","serif";} Qualitative robustness, influence function, and breakdown point are three main concepts to judge an estimator from the viewpoint of robust estimation. It is important as well as interesting to study relation among them. This article attempts to present the concept of qualitative robustness as forwarded by first proponents and its later development. It illustrates intricacies of qualitative robustness and its relation with consistency, and also tries to remove commonly believed misunderstandings about relation between influence function and qualitative robustness citing some examples from literature and providing a new counter-example. At the end it places a useful finite and a simulated version of qualitative robustness index (QRI). In order to assess the performance of the proposed measures, we have compared fifteen estimators of correlation coefficient using simulated as well as real data sets.
Highlights
Hampel in his Ph.D. thesis (1968) developed three concepts: qualitative robustness ( -robustness), breakdown point and influence function to assess robustness in estimation and raised rigorousness in robust estimation to a satisfactory level
We have demonstrated in the first proposition of the section the same holds for general c.s.m.s. b) Tn is q.r at F. and consistent at G in a nbd of F T is continuous at F
The following new counter example shows that even two-valued almost constant influence function does not guarantee the weak continuity of the functional; Let T( F ) = size of the largest atom and F0 y
Summary
Hampel in his Ph.D. thesis (1968) developed three concepts: qualitative robustness ( -robustness), breakdown point and influence function to assess robustness in estimation and raised rigorousness in robust estimation to a satisfactory level. Cuevas (1987 and 1988) adjusted some results of Hampel (1971) and Huber (1981) in the context of abstract inference He showed incompatibility of consistency and qualitative robustness in the case of kernel density estimators.
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