Deficiency (statistics) explained

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

l{E}

in relation to

l{F}

is zero, then one says

l{E}

is better or more informative or stronger than

l{F}

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

\Theta

as a triple

(X,E,(P_\theta)_{\theta\in\Theta})

consisting of a set

, a vector lattice

with unit

and a family of normalized positive functionals

(P_\theta)_\theta

.^[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

\Theta

be a parameter space. Given an abstract L₁-space

(L,\| ⋅ \|)

(i.e. a Banach lattice such that for elements

x,y\geq0

also

\|x+y\|=\|x\|+\|y\|

holds) consisting of lineare positive functionals

\{P_\theta:\theta\in\Theta\}

. An experiment

l{E}

is a map

l{E}:\Theta\toL

of the form

\theta\mapstoP_\theta

, such that

\|P_\theta\|=1

is the band induced by

\{P_\theta:\theta\in\Theta\}

and therefore we use the notation

L(l{E})

. For a

\mu\inL(l{E})

denote the

\mu⁺=\mu\vee0=max(\mu,0)

. The topological dual

of an L-space with the conjugated norm

\|u\|_{M=\sup\{|\langle}u,\mu\rangle|;\|\mu\|_L\leq1\}

is called an abstract M-space. It's also a lattice with unit defined through

	-\\|
\mu=\\|\mu
	L

for

\mu\inL

Let

L(A)

and

L(B)

be two L-space of two experiments

and

, then one calls a positive, norm-preserving linear map, i.e.

\|T\mu⁺\|=\|\mu⁺\|

for all

\mu\inL(A)

, a transition. The adjoint of a transitions is a positive linear map from the dual space

M_B

L(B)

into the dual space

M_A

L(A)

, such that the unit of

M_A

is the image of the unit of

M_B

ist.

Deficiency

Let

\Theta

be a parameter space and

l{E}:\theta\toP_\theta

and

l{F}:\theta\toQ_\theta

be two experiments indexed by

\Theta

. Le

L(l{E})

and

L(l{F})

denote the corresponding L-spaces and let

l{T}

be the set of all transitions from

L(l{E})

L(l{F})

The deficiency

\delta(l{E},l{F})

l{E}

in relation to

l{F}

is the number defined in terms of inf sup:

\delta(l{E},l{F}):=inf\limits_T\in

}\sup\limits_ \tfrac\|Q_-TP_\|_,^[6] where

\| ⋅ \|_TV

denoted the total variation norm

\|\mu\|_TV=\mu⁺+\mu^-

. The factor

\tfrac{1}{2}

is just for computational purposes and is sometimes omitted.

Le Cam distance

The Le Cam distance is the following pseudometric

\Delta(l{E},l{F}):=\operatorname{max}\left(\delta(l{E},l{F}),\delta(l{F},l{E})\right).

This induces an equivalence relation and when

\Delta(l{E},l{F})=0

, then one says

l{E}

and

l{F}

are equivalent. The equivalent class

C_l{E

} of

l{E}

is also called the type of
l{E}

.

Often one is interested in families of experiments

(l{E}_n)_n

with

\{P_n,\theta\colon\theta\in\Theta_n\}

and

(l{F}_n)_n

with

\{Q_n,\theta\colon\theta\in\Theta_n\}

. If

\Delta(l{E}_n,l{F}_n)=0

n\toinfty

, then one says

(l{E}_n)

and

(l{F}_n)

are asymptotically equivalent.

Let

\Theta

be a parameter space and

E(\Theta)

be the set of all types that are induced by

\Theta

, then the Le Cam distance

\Delta

is complete with respect to

E(\Theta)

. The condition

\delta(l{E},l{F})=0

induces a partial order on

E(\Theta)

, one says

l{E}

is better or more informative or stronger than

l{F}

References

Lucien . Le Cam . Sufficiency and Approximate Sufficiency . 35 . . 4 . . 1429 . 1964 . 10.1214/aoms/1177700372 . free .
Book: Torgersen, Erik . Comparison of Statistical Experiments. Cambridge University Press, United Kingdom . 1991 . 10.1017/CBO9780511666353 . 222-257.
Lucien . Le Cam . Sufficiency and Approximate Sufficiency . 35 . . 4 . . 1421 . 1964 . 10.1214/aoms/1177700372 . free .
Aad . van der Vaart . The Statistical Work of Lucien Le Cam . The Annals of Statistics . 30 . 3 . 2002 . 631–82 . 2699973.
Book: Le Cam, Lucien . Asymptotic methods in statistical decision theory . Springer Series in Statistics . Springer, New York. 1986 . 1-5 . 10.1007/978-1-4612-4946-7.
Book: Le Cam, Lucien . 1986 . 10.1007/978-1-4612-4946-7 . Springer, New York . 18-19 . Springer Series in Statistics . Asymptotic methods in statistical decision theory.

Bibliography

Book: Le Cam , Lucien . Asymptotic methods in statistical decision theory . Springer Series in Statistics . Springer, New York . 1986 . 10.1007/978-1-4612-4946-7.
Lucien . Le Cam . Sufficiency and Approximate Sufficiency . 35 . . 4 . . 1419 - 1455 . 1964 . 10.1214/aoms/1177700372. free .
Book: Torgersen , Erik . Comparison of Statistical Experiments . Cambridge University Press, United Kingdom . 1991 . 10.1017/CBO9780511666353.