Statistical and financial applications of constrained Dantzig-type estimators

Abstract

Traditional optimisation techniques face significant challenges in handling high-dimensional data with limited and noisy information. This paper explores the use of the Constrained Dantzig-type Estimator (CDE) in two distinct scenarios. First, to optimise website advertising by adapting a technique previously applied in sparse portfolio construction to the domain of online advertising. Unlike prior methods such as Constrained Lasso (CLasso), CDE allows the problem to be formulated as a Linear Programming Problem (LP) instead of a Quadratic one, offering substantial computational advantages in high-dimensional settings while encouraging sparsity in its solutions. In a high-dimensional setting, where the sample covariance is often singular, we explore three different formulations of CDE to estimate the precision matrix as an alternative to existing methods to produce a sparse and positive-definite solution while directly incorporating symmetric conditions into the optimisation problem, comparing their performances to a previously tested benchmark, the Constrained $\ell_1$-Minimization for Inverse Matrix Estimation (CLIME). Finally, we provide a theoretical analysis of CDE V3 by recasting the \(p\times p\) matrix problem as a single \(p^2\)-dimensional CDE. Under mild regularity, sparsity, and restricted eigenvalue (RE) conditions, we prove non-asymptotic \(\ell_1\) and Frobenius-norm error bounds of order \(\mathcal O(s\,\lambda/\kappa(s)^2)\), where \(s\) is the number of nonzeros and \(\kappa\) is the RE constant.