  Extreme quantiles are critical for understanding the behavior of data in the
tail region of a distribution. It is challenging to estimate extreme quantiles,
particularly when dealing with limited data in the tail. In such cases, extreme
value theory offers a solution by approximating the tail distribution using the
Generalized Pareto Distribution (GPD). This allows for the extrapolation beyond
the range of observed data, making it a valuable tool for various applications.
However, when it comes to conditional cases, where estimation relies on
covariates, existing methods may require computationally expensive GPD fitting
for different observations. This computational burden becomes even more
problematic as the volume of observations increases, sometimes approaching
infinity. To address this issue, we propose an interpolation-based algorithm
named EMI. EMI facilitates the online prediction of extreme conditional
quantiles with finite offline observations. Combining quantile regression and
GPD-based extrapolation, EMI formulates as a bilevel programming problem,
efficiently solvable using classic optimization methods. Once estimates for
offline observations are obtained, EMI employs B-spline interpolation for
covariate-dependent variables, enabling estimation for online observations with
finite GPD fitting. Simulations and real data analysis demonstrate the
effectiveness of EMI across various scenarios.
