Estimation of the sub-Gaussian Parameter

The sub-Gaussian parameter (also called the variance proxy) of a mean-zero random variable X is defined as xi2_star = supλ ∈ ℝ L(λ) where L(λ) = frac2λ2 log mathbbE eλ X is a weighted cumulant generating function. We study the estimation of xi2_star and prove that the minimax risk is governed by a non-increasing function δP(C) = sup|λ| ≥ C L(λ) - sup|λ| ≤ C L(λ) which captures the influence of the tail behavior of the distribution P. Over the class of distributions with δP ≤ r for a non-increasing function r, the minimax risk is, up to a multiplicative constant, lower bounded by r(√log n) + n-1/2 and upper bounded by r((log n)1/2-varepsilon) + n-1/2 + varepsilon for any varepsilon > 0. Our estimator for the upper bound is based on constrained maximization of the empirical analogue of L. In addition to being almost minimax optimal and adaptive, we further prove that the estimator is asymptotic normal under suitable conditions and that, if the underlying distribution is not sub-Gaussian, the estimator diverges with a rate determined by the heaviness of the distributional tail.