 Empirical likelihood

Empirical likelihood assigns adjustable probability masses to observed values, rather than selecting a fixed finite-dimensional probability distribution family. For distinct observations $x_1,\dots ,x_n$ from an independent and identically distributed sample, the basic objective is $L(p)={\prod }_i{p}_i$ with $p_i\ge 0$ and ${\sum }_i{p}_i=1$. Without additional constraints the maximum is $n^{-n}$ at $p_i=1/n$, recovering the empirical distribution.

To assess a proposed expected value $\mu$, impose ${\sum }_i{p}_i(x_i-\mu )=0$ and maximize under that constraint. The empirical likelihood ratio isFor observations $0,2$ and proposed expected value $1/2$, the constraints force weights $(3/4,1/4)$, giving $R=3/4$ and $-2\log R=2\log (4/3)$. A proposed expected value outside the convex hull of the observations has no feasible weights. Such sample-support restrictions matter in small samples and should not be confused with proof that the population expected value is impossible. The method's likelihood-without-a-parametric-family motivation is described by Art B. Owen.^[ref]