12.5 The Empirical Likelihood concept
12.5.1 Introduction into Empirical Likelihood
Let us now as in Owen (1988) and Owen (1990)
introduce the empirical likelihood (EL) concept.
Suppose a sample
of independent identically distributed
random variables in
according to a probability law with
unknown distribution function
and unknown density
.
For an observation
of
the likelihood function is given by
 |
(12.8) |
The empirical density calculated from the observations
is
 |
(12.9) |
where
denotes the indicator function.
It is easy to see that
maximizes
in the class of all
probability density functions.
The objective of the empirical likelihood concept is the construction
of tests and confidence intervals for a parameter
of
the distribution of
. To keep things simple we illustrate the
empirical likelihood method
for the expectation
. The null hypothesis is
.
We can test this assumption based on the empirical likelihood ratio
 |
(12.10) |
where
maximizes
subject to
 |
(12.11) |
On a heuristic level we can reject the null hypothesis
``under the true distribution
,
has expectation
''
if the ratio
is small
relative to
, i.e. the test rejects if
for a certain
level
. More precisely, Owen (1990)
proves the following
THEOREM 12.1
Let

be iid one-dimensional random variables
with expectation

and variance

. For a positive

let
be the set of all possible expectations of

with respect to
distributions

dominated by

(

). Then it follows
![$\displaystyle \lim_{n \rightarrow \infty} \textrm{P}[ \theta \in C_{r,n} ]
= \textrm{P}[ \chi^2 \leq -2 \log r ]$](xfghtmlimg2041.gif) |
|
|
(12.12) |
where

is a

-distributed random variable with
one degree of freedom.
From Theorem 12.1 it follows directly
This result suggests therefore to use the log-EL ratio
as the basic element of a test about a parametric hypothesis for
the drift function of a diffusion process.
12.5.2 Empirical Likelihood for Time Series Data
We will now expand the results in Section 12.5.1 to the case of
time series data. For an arbitrary
and any function
we have
![$\displaystyle \textrm{E}\left[ K\left({x-X_i\over h}\right) \{Y_i-\mu(x)\} \; \Big\vert \; \textrm{E}[Y_i\vert X_i=x] = \mu(x) \right] = 0 .$](xfghtmlimg2044.gif) |
(12.13) |
Let
be nonnegative numbers representing a density for
The empirical likelihood for
is
 |
(12.14) |
subject to
and
.
The second condition reflects (12.13).
We find the maximum
by introducing Lagrange multipliers and maximizing the Lagrangian
function
The partial derivatives are
With
we obtain as a solution to (12.14) the optimal weights
![$\displaystyle p_i(x) = n^{-1} \left[ 1 + \lambda(x) K\left({x-X_i\over h}\right) \{ Y_i-\mu(x)\} \right]^{-1}$](xfghtmlimg2056.gif) |
(12.15) |
where
is the root of
 |
(12.16) |
Note, that
follows from
The maximum empirical likelihood is achieved at
corresponding
to the nonparametric curve estimate
.
For a parameter estimate
we get the maximum empirical likelihood
for the smoothed parametric model
.
The log-EL ratio is
To study properties of the empirical likelihood based test statistic we need to evaluate
at an arbitrary
first, which requires the
following lemma on
that is proved in Chen et al. (2001).
Let
be a random process with
.
Throughout this chapter we use the notation
(
)
to denote the facts that
(
) for a sequence
.
Let
for
.
An application of the power series expansion of
applied to
(12.16) and Lemma 12.1 yields
Inverting the above expansion, we have
 |
(12.17) |
From (12.15), Lemma 12.1 and the Taylor expansion
of
we get
Inserting (12.17) in (12.18) yields
 |
(12.18) |
For any
, let
be the variance and the bias coefficient functions associated
with the NW estimator, respectively, see Wand and Jones (1995).
Let
For
,
converges to
the set of interior points in
. If
, we have
and
.
Define
Clearly,
is the asymptotic variance of
when
which is one of the
conditions we assumed.
It was shown by Chen et al. (2001), that
In the same paper it is shown, that condition (iii) entails
.
These and (12.19) mean that
Therefore,
is asymptotically equivalent to a studentized
distance
between
and
.
It is this property that leads us to use
as the basic building block in the construction of a global test statistic
for distinction between
and
in the next section.
The use of the empirical likelihood as a distance
measure and its comparison with other distance measures have been discussed in
Owen (1991) and Baggerly (1998).