  Consider a random sample from a bivariate distribution function $F$ in the
max-domain of attraction of an extreme-value distribution function $G$. This
$G$ is characterized by two extreme-value indices and a spectral measure, the
latter determining the tail dependence structure of $F$. A major issue in
multivariate extreme-value theory is the estimation of the spectral measure
$\Phi_p$ with respect to the $L_p$ norm. For every $p\in[1,\infty]$, a
nonparametric maximum empirical likelihood estimator is proposed for $\Phi_p$.
The main novelty is that these estimators are guaranteed to satisfy the moment
constraints by which spectral measures are characterized. Asymptotic normality
of the estimators is proved under conditions that allow for tail independence.
Moreover, the conditions are easily verifiable as we demonstrate through a
number of theoretical examples. A simulation study shows a substantially
improved performance of the new estimators. Two case studies illustrate how to
implement the methods in practice.
