Theory of Statistics - George Mason University

578 8 Nonparametric and Robust Inference
and
pH(y) =
m
k=1
nk
nvk
As we have noted already, this is a bona fide estimator:
IR d
pH(y) ≥ 0
pH(y)dy =
m
k=1
= 1.
ITk(y). (8.33)
nk
vk
Although our discussion generally concerns observations on multivariate
random variables, we should occasionally consider simple univariate observations.
One reason why the univariate case is simpler is that the derivative is a
scalar function. Another reason why we use the univariate case as a model is
because it is easier to visualize. The density of a univariate random variable
is two-dimensional, and densities of other types of random variables are of
higher dimension, so only in the univariate case can the density estimates be
graphed directly.
In the univariate case, we assume that the support is the finite interval
[a, b]. We partition [a, b] into a grid of m nonoverlapping bins Tk = [tn,k, tn,k+1)
where
a = tn,1 < tn,2 < . . . < tn,m+1 = b.
The univariate histogram is
pH(y) =
m
k=1
nvk
nk
n(tn,k+1 − tn,k) ITk(y). (8.34)
If the bins are of equal width, say h (that is, tk = tk−1+h), the histogram
is
pH(y) = nk
, for y ∈ Tk.
nh
This class of functions consists of polynomial splines of degree 0 with fixed
knots, and the histogram is the maximum likelihood estimator over the class
of step functions. Generalized versions of the histogram can be defined with
respect to splines of higher degree. Splines with degree higher than 1 may
yield negative estimators, but such histograms are also maximum likelihood
estimators over those classes of functions.
The histogram as we have defined it is sometimes called a “density histogram”,
whereas a “frequency histogram” is not normalized by the n.
Theory of Statistics c○2000–2013 James E. Gentle