LikelihoodRatioTest¶
- LikelihoodRatioTest(model0NbParameters, model0LogLikelihood, model1NbParameters, model1LogLikelihood, level=0.05)¶
Nested likelihood model selection.
This test helps selecting between two nested models
estimated by likelihood maximization.
Suppose that
is a model with
-dimensional parameter
where
is a
-dimensional subset of
. We suppose that
is obtained by constraining
to be equal to a fixed value denoted by
. Then
is a model with parameter
.
Let
denote the maximum likelihood estimator of
for model
and
its maximized log-likelihood.
The profile log-likelihood for
is defined as:
We define the profile deviance statistic depending on
as the maximized log-likelihood with respect to
when
is frozen:
Under suitable regularity conditions, for large
,
follows a
distribution.
We use the profile deviance statistic to define the
confidence region for the true value of parameter
:
where
is the
quantile of the
distribution. The level
is the Type 1 error of the test, which means the mistaken rejection of model
.
In order to test the validity of model
with parameter
relative to
with parameter
at the
-level of significance, we check the value
:
if
, we accept
rather than
,
if
, we reject
in favor of
.
- Parameters:
- model0NbParametersint,
,
Number of parameters of
- m0llhfloat
log-likelihood
- model1NbParametersint,
,
Number of parameters of
- m1llhfloat
log-likelihood
- levelfloat,
, optional
Risk of wrongly rejecting
Default value is 0.05
- model0NbParametersint,
- Returns:
- test_result
TestResult
Test result.
- test_result
Notes
Both models must have been estimated on the same data.