Employing the MCMC technique to compute the projection depth in high dimensions

Abstract

Projection depth (PD), one of the prevailing location depth notions, is a powerful and favored tool for multivariate nonparametric analysis. It permits the extension of the univariate median and weighted means to a multivariate setting. The multidimensional projection depth median (PM) and depth weighted means, including the Stahel–Donoho (SD) estimator are highly robust and affine equivariant. PM has the highest finite sample breakdown point robustness among affine equivariant location estimators.However, the computation of PD remains a challenge because its exact computation is only feasible for a data set with a dimension that is theoretically no higher than eight but practically no higher than three. Approximate algorithms such as random direction procedure or simulated annealing (SA) algorithm, are time-consuming in high-dimensional cases. Here, we present an efficient SA algorithm and its extension for the computation of PD. Simulated and real data examples indicate that the proposed algorithms outperform their competitors, including the Nelder–Mead method, and the SA algorithm, in high-dimensional cases and can obtain highly accurate results compared with those of the exact algorithm in low-dimensional cases.