In statistics, the deficiency is a measure to compare a statistical model with another statistical model.
The concept was introduced in the 1960s by the French mathematician Lucien Le Cam, who used it to prove an approximative version of the Blackwell–Sherman–Stein theorem.
[1][2] Closely related is the Le Cam distance, a pseudometric for the maximum deficiency between two statistical models.
If the deficiency of a model
is better or more informative or stronger than
Le Cam defined the statistical model more abstract than a probability space with a family of probability measures.
He also didn't use the term "statistical model" and instead used the term "experiment".
In his publication from 1964 he introduced the statistical experiment to a parameter set
and a family of normalized positive functionals
[5] This article follows his definition from 1986 and uses his terminology to emphasize the generalization.
be a parameter space.
(i.e. a Banach lattice such that for elements
holds) consisting of lineare positive functionals
of an L-space with the conjugated norm
is called an abstract M-space.
It's also a lattice with unit defined through
be two L-space of two experiments
, then one calls a positive, norm-preserving linear map, i.e.
The adjoint of a transitions is a positive linear map from the dual space
be a parameter space and
be the set of all transitions from
is the number defined in terms of inf sup: where
denoted the total variation norm
is just for computational purposes and is sometimes omitted.
The Le Cam distance is the following pseudometric This induces an equivalence relation and when
Often one is interested in families of experiments
be the set of all types that are induced by
, then the Le Cam distance
is complete with respect to
induces a partial order on