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 in relation to is zero, then one says is better or more informative or stronger than .
Introduction
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 as a triple consisting of a set , a vector lattice with unit and a family of normalized positive functionals on .[3][4] In his book from 1986 he omitted and .[5] This article follows his definition from 1986 and uses his terminology to emphasize the generalization.
Formulation
Basic concepts
Let be a parameter space. Given an abstract L1-space (i.e. a Banach lattice such that for elements also holds) consisting of lineare positive functionals . An experiment is a map of the form , such that . is the band induced by and therefore we use the notation . For a denote the . The topological dual of an L-space with the conjugated norm is called an abstract M-space. It's also a lattice with unit defined through for .
Let and be two L-space of two experiments and , then one calls a positive, norm-preserving linear map, i.e. for all , a transition. The adjoint of a transitions is a positive linear map from the dual space of into the dual space of , such that the unit of is the image of the unit of ist.[5]
Deficiency
Let be a parameter space and and be two experiments indexed by . Le and denote the corresponding L-spaces and let be the set of all transitions from to .
The deficiency of in relation to is the number defined in terms of inf sup:
where denoted the total variation norm . The factor is just for computational purposes and is sometimes omitted.
Le Cam distance
The Le Cam distance is the following pseudometric
This induces an equivalence relation and when , then one says and are equivalent. The equivalent class of is also called the type of .
Often one is interested in families of experiments with and with . If as , then one says and are asymptotically equivalent.
Let be a parameter space and be the set of all types that are induced by , then the Le Cam distance is complete with respect to . The condition induces a partial order on , one says is better or more informative or stronger than .[6]
References
- ^ Le Cam, Lucien (1964). "Sufficiency and Approximate Sufficiency". Annals of Mathematical Statistics. 35 (4). Institute of Mathematical Statistics: 1429. doi:10.1214/aoms/1177700372.
- ^ Torgersen, Erik (1991). Comparison of Statistical Experiments. Cambridge University Press, United Kingdom. pp. 222–257. doi:10.1017/CBO9780511666353.
- ^ Le Cam, Lucien (1964). "Sufficiency and Approximate Sufficiency". Annals of Mathematical Statistics. 35 (4). Institute of Mathematical Statistics: 1421. doi:10.1214/aoms/1177700372.
- ^ van der Vaart, Aad (2002). "The Statistical Work of Lucien Le Cam". The Annals of Statistics. 30 (3): 631–82. JSTOR 2699973.
- ^ a b Le Cam, Lucien (1986). Asymptotic methods in statistical decision theory. Springer Series in Statistics. Springer, New York. pp. 1–5. doi:10.1007/978-1-4612-4946-7.
- ^ a b Le Cam, Lucien (1986). Asymptotic methods in statistical decision theory. Springer Series in Statistics. Springer, New York. pp. 18–19. doi:10.1007/978-1-4612-4946-7.
Bibliography
- Le Cam, Lucien (1986). Asymptotic methods in statistical decision theory. Springer Series in Statistics. Springer, New York. doi:10.1007/978-1-4612-4946-7.
- Le Cam, Lucien (1964). "Sufficiency and Approximate Sufficiency". Annals of Mathematical Statistics. 35 (4). Institute of Mathematical Statistics: 1419–1455. doi:10.1214/aoms/1177700372.
- Torgersen, Erik (1991). Comparison of Statistical Experiments. Cambridge University Press, United Kingdom. doi:10.1017/CBO9780511666353.