Selected Glossary for Robust Statistics
(and Statistics, Generally)

Functional -- this is a convenient word that refers to
  an operation on a function (such as a CDF), on
  a population, or on a sample.   For example, the
  mean: informally we may speak of a mean as 
  1. a functional as an integral with
      respect to the CDF, 
  2. an expected value of a random variable (which is a 
      function), 
  3. a parameter ("measure") of the population,
  4. a statistic computed from a sample

Invariance (of a functional) -- the functional is unaffected
  by the indicated type of transformation on the argument of
  the functional.  For example, the variance functional is
  unaffected by the addition of a constant to each observation
  in a sample.  The phrase describing this is "location
  invariance of the variance" or "translation invariance of
  the variance".

Equivariance (of a functional) -- the functional is changed
  by the same transformation on the argument of the functional.
  For example, the result of the mean functional is increased
  by the same amount as that of a constant added to each
  observation in a sample.  Likewise, the result of the mean
  functional is multiplied by the same amount as that of a 
  constant by which each observation in a sample is multiplied.
  We therefore say the "mean is location equivariant and is
  scale equivariant".  A special variant of scale equivariance
  is absolute scale equivariance, in which the functional is
  even, but otherwise is scale equivariant.

Robustness -- This term is usually applied to a statistical
  procedure, and it means generally that the procedure is
  not greatly affected by failures of the assumptions.
  The term is also applied in a formal way to functionals.
  There are three specific, but not too-well-thought-out 
  aspects of this concept.  The distinctions are of rather
  limited utility, but they are commonly defined in textbooks
  and survey articles on robustness.  Two types of "robustness"
  are dicotomous properties, and one type of "robustness" is
  measured by a real number from 0 to 1/2.  One type of
  "robustness" is a property of a functional (only), and
  two types are properties of a functional and a distribution. 
  
Qualitative robustness of a functional -- the functional is
  continuous.  This concept is somewhat vague because the
  meaning of continuity (in terms of epsilons and deltas) is
  not specified in defining this type of robustness.

Infinitesimal robustness of a functional (T) of a CDF (P) -- 
  the influence function (T, P) is bounded.

Quantitative robustness of a functional (T) of a CDF (P) -- 
  the sup value of epsilon in the epsilon-mixture for which
  the difference in the functional applied to the epsilon-mixture
  and the function applied to P is bounded.

For robustness of estimators, we generally distinguish 
"location" estimators and "scale estimators". 

Location estimator -- a linear funtional that has the same
sign as the random variable.
A linear is location and scale equivariant.

Scale estimator -- a nonnegative funtional that is location 
invariant and absolutely scale equivariant. 

Some interesting location estimators:
 * Winsorized means (proportion gamma)
 * trimmed means (proportion gamma)
 * M-estimators of location (rho, or xi, and psi functions)
 * R-estimators of location (ranks)
 * L-estimators of location (order statistics; 
    these include median and other quantile estimators)
 * variations of M-estimators defined by an approximation or by a 
    computational method: one-step M-estimator, and "W-estimator",
    which is computed by iterative reweighting.

M-estimators deserve special discussion.  They are based on a distance
measure rho (which Wilcox calls xi).  They are defined as the value
of m that minimizes
  sum (rho(x_i-m))
Usually rho is differentiable, so the first-order minimization property
requires that 
  sum (psi(x_i-m))=0.
The influence function for an M-estimator can be quite complicated.
For simple values of rho, however, the influence function can be shown
to be psi(x-m)/E(psi'(X-m)).  Some special rhos and corresponding psis
are called Huber, Andrews, Hampel, and biweight.

M-estimators are usually defined so as to be asymptotically scale
equivariant, so the definition is modified to be the m that minimizes
  sum (rho((x_i-m)/tau)),
where tau is a consistent (and independent!) estimator of the scale.

Some interesting scale estimators:
 * mean absolute deviation from the mean
 * mean absolute deviation from the median
 * median absolute deviation from the median (MAD)
 * normalized MAD, MADN (this is the S-Plus function mad)
 * interquantile range
 * Winsorized variance
 * biweight midvariance (Wilcox, p. 62)
 * percentage bend midvariance (Wilcox, p. 64)

Efficiency (w.r.t. an MVUE)
Triefficiency (normal, one-wild, and slash distributions)
Asymptotic relative efficiency (ARE)

Bootstrap
Percentile bootstrap
Percentile t bootstrap
Modified, symmetric percentile t bootstrap