This research introduces a Q-Q orthogonality formulation to separate the causes of instability in sample quantiles derived from heavy-tailed distributions, specifically addressing the effects of projection direction and quantile thresholds.
The study examines the stability and decomposition of sample quantiles relevant to Value-at-Risk (VaR) projections in financial returns that follow heavy-tailed probability laws. While empirical-process theory provides stability bounds, the paper develops a new Q-Q orthogonality framework to disentangle the instability arising from two distinct factors: the direction of projection and the quantile threshold. The work decomposes the difference between empirically computed and population quantiles into three terms ($D_1, D_2, D_3$), allowing for a more precise understanding of how these different sources of instability manifest.