  Data from the James Webb Space Telescope have revealed an intriguing
population of bright galaxies at high redshifts. In this work, we use
extreme-value statistics to calculate the distribution (in UV magnitude) of the
brightest galaxies in the redshift range $9 \lesssim z \lesssim 16$. We combine
the Generalised Extreme Value (GEV) approach with modelling of the galaxy
luminosity function. We obtain predictions of the brightest galaxies for a
suite of luminosity functions, including the Schechter and double power law
functions, as well as a model parametrised by the stellar formation efficiency
$f_*$. We find that the \textit{JWST} data is broadly consistent with $f_*$ of
$5\%-10\%$, and that the brightest galaxy at $z\sim16$ will have $M_{\rm
UV}\approx -23.5^{0.8}_{0.4}$. If $f_*$ is dependent on halo mass, we predict
$M_{\rm UV}\approx -22.5^{0.5}_{1.5}$ for such an object. We show that
extreme-value statistics not only predicts the magnitude of the brightest
galaxies at high redshifts, but may also be able to distinguish between models
of star formation in high-redshift galaxies.
