Empirical likelihood

Given a set of ${\displaystyle n}$ i.i.d. realizations ${\displaystyle y_{i}}$ of random variables ${\displaystyle Y_{i}}$, then the empirical distribution function is ${\displaystyle {\hat {F}}(y):=\sum _{i=1}^{n}\pi _{i}I(Y_{i}<y)}$, with the indicator function ${\displaystyle I}$ and the (normalized) weights ${\displaystyle \pi _{i}}$. Then, the empirical likelihood is:[3]

${\displaystyle L:=\prod _{i=1}^{n}{\frac {{\hat {F}}(y_{i})-{\hat {F}}(y_{i}-\delta y)}{\delta y}},}$

where ${\displaystyle \delta y}$ is a small number (potentially the difference to the next smaller sample).

Important is the that the 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 ${\displaystyle E[h(Y;\theta )]=\int _{-\infty }^{\infty }h(y;\theta )dF=0}$ which implies ${\displaystyle {\hat {E}}[h(y;\theta )]=\sum _{i=1}^{n}h(y_{i};\theta )\pi _{i}=0}$.

With similar constraints, we could also model correlation.

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:

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

subject to the constraints

${\displaystyle s.t.\sum _{i=1}^{n}\pi _{i}=1,\sum _{i=1}^{n}\pi _{i}h(y_{i};\theta )=0,\forall i\in [1..n]\quad 0\leq \pi _{i}.}$[6]^{: Equation (73)}

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

${\displaystyle {\mathcal {L}}=\sum _{i=1}^{n}\ln \pi _{i}+\mu (1-\sum _{i=1}^{n}\pi _{i})-n\tau '\sum _{i=1}^{n}\pi _{i}h(y_{i};\theta ).}$[6]^{: Equation (74)}

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

The parameters ${\displaystyle \pi _{i}}$ are nuisance parameters.

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 ${\displaystyle F}$, then the ELR would be:

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

Consider sets of the form

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

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

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

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

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

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

• Bootstrapping (statistics)
• Jackknife (statistics)
• Nelson–Aalen estimator

• Nordman, Daniel J., and Soumendra N. Lahiri. "A review of empirical likelihood methods for time series." Journal of Statistical Planning and Inference 155 (2014): 1-18. https://doi.org/10.1016/j.jspi.2013.10.001