In probability theory and statistics, empirical likelihood (EL) is a nonparametric method for estimating the parameters of statistical models.
It requires fewer assumptions about the error distribution while retaining some of the merits in likelihood-based inference.
The estimation method requires that the data are independent and identically distributed (iid).
It performs well even when the distribution is asymmetric or censored.
[1] EL methods can also handle constraints and prior information on parameters.
Art Owen pioneered work in this area with his 1988 paper.
, then the empirical distribution function is
, with the indicator function
and the (normalized) weights
Then, the empirical likelihood is:[3] where
is a small number (potentially the difference to the next smaller sample).
Empirical likelihood estimation can be augmented with side information by using further constraints (similar to the generalized estimating equations approach) for the empirical distribution function.
E.g. a constraint like the following can be incorporated using a Lagrange multiplier
With similar constraints, we could also model correlation.
The empirical-likelihood method can also be also employed for discrete distributions.
Then the empirical likelihood is again
Using the Lagrangian multiplier method to maximize the logarithm of the empirical likelihood subject to the trivial normalization constraint, we find
is the empirical distribution function.
EL estimates are calculated by maximizing the empirical likelihood function (see above) subject to constraints based on the estimating function and the trivial assumption that the probability weights of the likelihood function sum to 1.
[5] This procedure is represented as: subject to the constraints The value of the theta parameter can be found by solving the Lagrangian function There is a clear analogy between this maximization problem and the one solved for maximum entropy.
An empirical likelihood ratio function is defined and used to obtain confidence intervals parameter of interest θ similar to parametric likelihood ratio confidence intervals.
[7][8] Let L(F) be the empirical likelihood of function
The problem can be solved by restricting to distributions F that are supported in a bounded set.
It turns out to be possible to restrict attention t distributions with support in the sample, in other words, to distribution
Such method is convenient since the statistician might not be willing to specify a bounded support for
into a finite dimensional problem.
The use of empirical likelihood is not limited to confidence intervals.
In efficient quantile regression, an EL-based categorization[9] procedure helps determine the shape of the true discrete distribution at level p, and also provides a way of formulating a consistent estimator.
In addition, EL can be used in place of parametric likelihood to form model selection criteria.
[10] Empirical likelihood can naturally be applied in survival analysis[11] or regression problems[12]