ABSTRACT

It is easy to see that empirical quantile estimates (including  Value-at-Risk estimates) can be very sensitive to small perturbations in the data. Traditionally, attempts to quantify the uncertainty in quantile estimates were based on construction of confidence intervals for quantile estimates, which in turn involves a density estimate appearing in a denominator. Such estimates suffer from several shortcomings, including the following:

(1) These estimates ignore the more rigorous definition of a quantile as an interval, and arbitrarily  picks one point in the interval (often the smallest point).

(2) If the empirical quantile is not a singleton, then a confidence interval for an arbitrarily picked point is a poor way to quantify the uncertainty of the quantile.

(3) The confidence interval estimates make sense for distributions with positive densities, but not for discrete
distributions, and unfortunately most real-world data, including data processed by a computer, have discrete distributions.

(4) Even for distributions with positive densities, the reliance on density estimates  can make the confidence interval estimates erratic. We propose an alternative approach to constructing robust quantile estimates, that
hedge against variation in the unknown distribution of the underlying random variables.

These distributionally robust quantile estimates have a number of desirable properties, such as uniqueness, and worst-case distributions with intuitive structure. We demonstrate the approach with evaluation of the Value-at-Risk of a portfolio.