  Existing theory for multivariate extreme values focuses upon
characterizations of the distributional tails when all components of a random
vector, standardized to identical margins, grow at the same rate. In this
paper, we consider the effect of allowing the components to grow at different
rates, and characterize the link between these marginal growth rates and the
multivariate tail probability decay rate. Our approach leads to a whole class
of univariate regular variation conditions, in place of the single but
multivariate regular variation conditions that underpin the current theories.
These conditions are indexed by a homogeneous function and an angular
dependence function, which, for asymptotically independent random vectors,
mirror the role played by the exponent measure and Pickands' dependence
function in classical multivariate extremes. We additionally offer an
inferential approach to joint survivor probability estimation. The key feature
of our methodology is that extreme set probabilities can be estimated by
extrapolating upon rays emanating from the origin when the margins of the
variables are exponential. This offers an appreciable improvement over existing
techniques where extrapolation in exponential margins is upon lines parallel to
the diagonal.
