Robust Statistics: definition
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Science Classics
Robust Statistics, 1981, New York, Wiley by Peter J. Huber.
Peter J. Huber
Peter J. Huber was born on March 25, 1934, in Wohlen, a small town in
the Swiss countryside. He obtained a diploma in mathematics in 1958 and a Ph.D.
in mathematics in 1961, both from ETH Zurich. His thesis was in pure
mathematics, but he then decided to go into statistics. He spent 1961–1963 as a
postdoc at the statistics department in Berkeley where he wrote his first and
most famous paper on robust statistics, “Robust Estimation of a Location
Parameter.” After a position as a visiting professor at Cornell University, he
became a full professor at ETH Zurich. He worked at ETH until 1978,
interspersed by visiting positions at Cornell, Yale, Princeton and Harvard.
After leaving ETH, he held professor positions at Harvard University 1978–1988,
at MIT 1988–1992, and finally at the University of Bayreuth from 1992 until his
retirement in 1999. He now lives in Klosters, a village in the Grisons in the
Swiss Alps.
Source: Statistical
Science
Robust statistics
Statistical
inferences are base only in part upon observations. An equally important base
is formed by prior assumptions about the underlying situation. Even in the
simplest cases, there are explicit or implicit assumptions about randomness and
independence, about distributional models, perhaps prior distributions for some unknown parameters, and so on.
These
assumptions are not supposed to be exactly true – they are mathematically
convenient rationalizations of an often fuzzy knowledge or belief. As in every
other branch of applied mathematics, such rationalizations or simplifications
are vital, and one justifies their use by appealing to a vague continuity or stability principle; a minor
error in the mathematical model should cause only a small error in the final
conclusions.
Unfortunately,
this does not always hold. During past
decades one has become increasingly aware that some of the most common
statistical procedures (in particular, those optimized for an underlying normal
distribution) are excessively sensitive
to seemingly minor deviations from the assumptions, and a plethora of
alternative “robust” procedures have
been proposed.
The word “robust“ is loaded with many
–sometimes-inconsistent – connotations. We use it in a relative narrow sense: for our purposes, robustness signifies
insensitivity to small deviations from the assumptions.
What should a robust procedure achieve?
Any
statistical procedure should possess the following desirable features:
- It should have a reasonably good (optimal or nearly optimal) efficiency at the assumed model. [Near-optimal calibration, ε-optimality, ε-subdifferential]
- It should be robust in the sense that small deviations from the model assumptions should impair the performance only slightly, that is, the latter (described, say, in terms of the asymptotic variance of an estimate, or of a level and power of a test) should be close to the nominal value calculated at the model. [Stability, continuity of ε-optimal set]
- Somewhat large deviations from the model should not cause a catastrophe.
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