Empirical likelihood

Empirical likelihood (EL) is a nonparametric method that 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. EL methods can also handle constraints and prior information on parameters. Art Owen pioneered work in this area with his 1988 paper.

Estimation Procedure

EL estimates are calculated by maximizing the empirical likelihood function subject to constraints based on the estimating function and the trivial assumption that the probability weights of the likelihood function sum to 1. This procedure is represented as:

$\max _{\pi _{i},\theta }\sum _{i=1}^{n}\ln \pi _{i}$

subject to the constraints

$s.t.\sum _{i=1}^{n}\pi _{i}=1,\sum _{i=1}^{n}\pi _{i}h(y_{i};\theta )=0,\forall i\in [n]\quad 0\leq \pi _{i}.$

The value of the theta parameter can be found by solving the Lagrangian function

${\mathcal {L}}=\sum _{i=1}^{n}\ln \pi _{i}+\mu (1-\sum _{i=1}^{n}\pi _{i})-n\tau '\sum _{i=1}^{n}\pi _{i}h(y_{i};\theta ).$

There is a clear analogy between this maximization problem and the one solved for maximum entropy.

The empirical-likelihood method can also be also employed for discrete distributions:

$F(x_{i})=\ p_{i},\ i=1,...,n$

where

$\ p_{i}\geq 0,\sum _{i=1}^{n}\ p_{i}=1.$

Then the likelihood $L(p_{1},...,p_{n})=\prod _{i=1}^{n}\ p_{i}$ is referred to as an empirical likelihood.

Empirical Likelihood Ratio (ELR)

An empirical likelihood ratio function is defined and used to obtain confidence intervals parameter of interest θ similar to parametric likelihood ratio confidence intervals. Let L(F) be the empirical likelihood of function $F$, then the ELR would be:

$R(F)=L(F)/L(F_{n})$.

Consider sets of the form

$C=\{T(F)|R(F)\geq r\}$.

Under such conditions a test of $T(F)=t$ rejects when t does not belong to $C$, that is, when no distribution F with $T(F)=t$ has likelihood $L(F)\geq rL(F_{n})$.

The central result is for the mean of X. Clearly, some restrictions on $F$ are needed, or else $C=\mathbb {R} ^{p}$ whenever $r<1$. To see this, let:

$F=\epsilon \delta _{x}+(1-\epsilon )F_{n}$

If $\epsilon$ is small enough and $\epsilon >0$, then $R(F)\geq r$.

But then, as $x$ ranges through $\mathbb {R} ^{p}$, so does the mean of $F$, tracing out $C=\mathbb {R} ^{p}$. 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 $F\ll F_{n}$. Such method is convenient since the statistician might not be willing to specify a bounded support for $F$, and since $t$ converts the construction of $C$ into a finite dimensional problem.

Other Applications

The use of empirical likelihood is not limited to confidence intervals. In quantile estimation, an EL-based categorization 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.

See also

• Bootstrapping (statistics)
• Jackknife (statistics)