Theory of Statistics - George Mason University

576 8 Nonparametric and Robust Inference
For a step function with m regions having constant values, c1, . . ., cm, the
likelihood is
L(c1, . . ., cm; y1, . . ., yn) =
=
n
p(yi)
i=1
m
k=1
c nk
k , (8.30)
where nk is the number of data points in the kth region. For the step function
to be a bona fide estimator, all ck must be nonnegative and finite. A maximum
therefore exists in the class of step functions that are bona fide estimators.
If vk is the measure of the volume of the kth region (that is, vk is the
length of an interval in the univariate case, the area in the bivariate case, and
so on), we have
m
ckvk = 1.
k=1
We incorporate this constraint together with equation (8.30) to form the Lagrangian,
m
L(c1, . . ., cm) + λ 1 −
.
k=1
ckvk
Differentiating the Lagrangian function and setting the derivative to zero, we
have at the maximum point ck = c∗ k , for any λ,
∂L
∂ck
= λvk.
Using the derivative of L from equation (8.30), we get
nkL = λc ∗ k vk.
Summing both sides of this equation over k, we have
and then substituting, we have
nL = λ,
nkL = nLc ∗ kvk.
Therefore, the maximum of the likelihood occurs at
c ∗ k = nk
.
nvk
The restricted maximum likelihood estimator is therefore
Theory of Statistics c○2000–2013 James E. Gentle