Theory of Statistics - George Mason University

752 0 Statistical Mathematics
0.1.13 Functionals
Functionals are functions whose arguments are functions. The value of a functional
may be any kind of object, a real number or another function, for
example. The domain of a functional is a set of functions.
If F is a linear space of functions, that is, if F is such that f ∈ F and
g ∈ F implies (af + g) ∈ F for any real a, then the functional Υ defined on
F is said to be linear if Υ(af + g) = aΥ(f) + Υ(g).
A similar expression defines linearity of a functional over a distribution
function space P: Υ defined on P is linear if Υ((1 − w)P1 + wP2) = (1 −
w)Υ(P1) + wΥ(P2) for w ∈ [0, 1] and P1, P2 ∈ P.
Functionals of CDFs have important uses in statistics as measures of the
differences between two distributions or to define distributional measures of
interest. A functional applied to a ECDF is a plug-in estimator of the distributional
measure defined by the same functional applied to the corresponding
CDF.
0.1.13.1 Derivatives of Functionals
For the case in which the arguments are functions, the cardinality of the
possible perturbations is greater than that of the continuum. We can be precise
in discussions of continuity and differentiability of a functional Υ at a point
(function) F in a domain F by defining another set D consisting of difference
functions over F; that is the set the functions D = F1 − F2 for F1, F2 ∈ F.
The concept of differentiability for functionals is necessarily more complicated
than for functions over real domains. For a functional Υ over the
domain F, we define three levels of differentiability at the function F ∈ F.
All definitions are in terms of a domain D of difference functions over F, and
a linear functional ΛF defined over D in a neighborhood of F. The first type
of derivative is very general. The other two types depend on a metric ρ on
F × F induced by a norm · on F.
Definition 0.1.53 (Gâteaux differentiable)
Υ is Gâteaux differentiable at F iff there exists a linear functional ΛF (D) over
D such that for t ∈ IR for which F + tD ∈ F,
Υ(F + tD) − Υ(F)
lim
− ΛF(D) = 0. (0.1.115)
t→0 t
In this case, the linear functional ΛF is called the Gâteaux differential of
Υ at F in the direction of F + D.
Definition 0.1.54 (ρ-Hadamard differentiable)
For a metric ρ induced by a norm, Υ is ρ-Hadamard differentiable at F iff
Theory of Statistics c○2000–2013 James E. Gentle