We investigate robust mean estimation through utilizing different loss functions, including the Huber, Catoni (Catoni, 2012), and two newly defined loss functions created to gauge the performance of over- or underestimating important ``spread'' parameters. We allow $E|x_i|^q < \infty$, where $1<q\leq 2$. This is a new contribution to the literature, when only the case of $q=2$ has been studied, i.e., the robust mean estimation with the existence of at least second order moments. This development is particularly important when we are estimating higher order moments themselves, like the kurtosis of financial return data. With our proposed robust mean estimator, we create a new eigenvalues-regularized covariance matrix estimator under the assumption $p/n \rightarrow c>0$, where $p$ is the dimension and $n$ is the sample size. This estimator needs only the existence of up to the $\xi$th order moments, $2<\xi\leq 4$, which is a huge relaxation compared to requiring the existence of 12th order moments in Lam (2016). Theoretical properties are analysed through a newly defined relative loss function, and extensive empirical studies and comparisons with contemporary estimators like the Huber-type M-estimator and rank-based estimator analyzed in Avella et al (2018) are performed, showing the superiority of our estimator. A set of ionosphere data is also analyzed in the end. 

Speaker: Professor Clifford LAM 
Date: 3 February 2021 (Wed)
Time: 4:00pm – 5:00pm
Poster: Click here