  The minimax risk is often considered as a gold standard against which we can
compare specific statistical procedures. Nevertheless, as has been observed
recently in robust and heavy-tailed estimation problems, the inherent reduction
of the (random) loss to its expectation may entail a significant loss of
information regarding its tail behaviour. In an attempt to avoid such a loss,
we introduce the notion of a minimax quantile, and seek to articulate its
dependence on the quantile level. To this end, we develop high-probability
variants of the classical Le Cam and Fano methods, as well as a technique to
convert local minimax risk lower bounds to lower bounds on minimax quantiles.
To illustrate the power of our framework, we deploy our techniques on several
examples, recovering recent results in robust mean estimation and stochastic
convex optimisation, as well as obtaining several new results in covariance
matrix estimation, sparse linear regression, nonparametric density estimation
and isotonic regression. Our overall goal is to argue that minimax quantiles
can provide a finer-grained understanding of the difficulty of statistical
problems, and that, in wide generality, lower bounds on these quantities can be
obtained via user-friendly tools.
