We study the problem of designing minimax procedures in linear regression under the quantile risk. We start by considering the realizable setting with independent Gaussian noise, where for any given noise level and distribution of inputs, we obtain the exact minimax quantile risk for a rich family of error functions and establish the minimaxity of OLS. This improves on the lower bounds obtained by Lecué and Mendelson (2016) and Mendelson (2017) for the special case of square error, and provides us with a lower bound on the minimax quantile risk over larger sets of distributions. Under the square error and a fourth moment assumption on the distribution of inputs, we show that this lower bound is tight over a larger class of problems. Specifically, we prove a matching upper bound on the worst-case quantile risk of a variant of the procedure proposed by Lecué and Lerasle (2020), thereby establishing its minimaxity, up to absolute constants. We illustrate the usefulness of our approach by extending this result to all p-th power error functions for p (2,). Along the way, we develop a generic analogue to the classical Bayesian method for lower bounding the minimax risk when working with the quantile risk, as well as a tight characterization of the quantiles of the smallest eigenvalue of the sample covariance matrix.

The optimality of allocating assets has been widely discussed with the theoretical analysis of risk measures. Pessimism is one of the most attractive approaches beyond the conventional optimal portfolio model, and the $\alpha$-risk plays a crucial role in deriving a broad class of pessimistic optimal portfolios. However, estimating an optimal portfolio assessed by a pessimistic risk is still challenging due to the absence of an available estimation model and a computational algorithm. In this study, we propose a version of integrated $\alpha$-risk called the uniform pessimistic risk and the computational algorithm to obtain an optimal portfolio based on the risk. Further, we investigate the theoretical properties of the proposed risk in view of three different approaches: multiple quantile regression, the proper scoring rule, and distributionally robust optimization. Also, the uniform pessimistic risk is applied to estimate the pessimistic optimal portfolio models for the Korean stock market and compare the result of the real data analysis. It is empirically confirmed that the proposed pessimistic portfolio presents a more robust performance than others when the stock market is unstable.