A simulation study of the finite-sample performance of the sample scale curve as an estimator of its population counterpart

The depth-based scale curves have broad applications in nonparametric multivariate statistics. Many nonparametric multivariate measures and procedures are based on the scale curves. In this article, we study the finite-sample performance of the sample scale curve as an estimator of its population counterpart by simulation. It is found that the sample scale curve tends to underestimate its population counterpart and that the discrepancy increases quickly as dimension increases. The effect of data shape characterized by kurtosis on the performance is also studied. Compared with the effect of dimension, the effect of kurtosis is minor. The root cause of the issue is explored and some suggestions for improvement are provided.