Theory of Statistics - George Mason University

738 0 Statistical Mathematics
A related pseudonorm is more useful, however, because it is the limit of
equation (0.1.75) as p → ∞ (compare equation (0.0.34)). We define
f∞ = ess sup |f(x)w(x)|, (0.1.77)
where ess sup denotes the essential supremum of a function, defined for a
given measure µ by
ess sup g(x) = inf{a : µ({x : x ∈ D, g(x) > a}) = 0}.
The essential infimum of a function for a given measure µ is defined similarly:
ess inf g(x) = sup{a : µ({x : x ∈ D, g(x) < a}) = 0}.
The pseudonorm defined by equation (0.1.77) is called the L∞ norm, the
Chebyshev norm, or the uniform norm.
Another type of function norm, called the total variation, is an L∞-type of
measure of the amount of variability of the function. For a real-valued scalar
function f on the interval [a, b], the total variation of f on [a, b] is
(f) = sup |f(xi+1) − f(xi)|, (0.1.78)
V b a
where π is a partition of [a, b], (a = x0 < x1 < · · · < xn = b).
If f is continuously differentiable over [a, b], then
π
V b a(f) =
i
b
a
|f ′ (x)|dx. (0.1.79)
A normal function is a function whose pseudonorm is 1. A normal function
is also called a normal function a normalized function. Although this term
can be used with respect to any pseudonorm, it is generally reserved for the
L2 pseudonorm (that is, the pseudonorm arising from the inner product). A
function whose integral (over a relevant range, usually IR) is 1 is also called
a normal function. (Although this latter definition is similar to the standard
one, the latter is broader because it may include functions that are not squareintegrable.)
Density and weight functions are often normalized (that is, scaled
to be normal).
0.1.9.4 Metrics in Function Spaces
Statistical properties such as bias and consistency are defined in terms of the
difference of the estimator and what is being estimated. For an estimator of a
function, first we must consider some ways of measuring this difference. These
are general measures for functions and are not dependent on the distribution
of a random variable. How well one function approximates another function is
Theory of Statistics c○2000–2013 James E. Gentle