A Coskewness Shrinkage Approach for Estimating the Skewness of Linear Combinations of Random Variables

Abstract

The coskewness matrix is a (p)x(p^2) parameter matrix useful for studying the third-order interactions between variables and computing the skewness of linear combinations of variables. The classical sample estimator of the coskewness matrix performs poorly in terms of mean squared error (MSE) when the sample size is small, even in moderate dimensions. A possible approach to improve estimation is to use shrinkage estimators that offer a compromise between the (unbiased) sample coskewness matrix and a (possibly biased) target matrix (with lower estimation variance) with the aim of minimizing the MSE. We improve upon the existing literature by proposing unbiased estimators, using multivariate k-statistics and polykays, for the MSE loss function and by including the possibility of having multiple target matrices. Simulations show the superiority of our estimators in both the single-target and the multi-target setting for skewed data. Accurate coskewness estimation is highly valued in the field of portfolio optimization (among others) where we show that the out-of-sample skewness of the maximum skewness optimized portfolio is higher when using the proposed shrinkage coskewness matrix estimates.