Correlation control in small sample Monte Carlo type simulations II: Analysis of estimation formulas, random correlation and perfect uncorrelatedness

This paper presents a number of theoretical and numerical results regarding correlation coefficients and two norms of correlation matrices in relation to correlation control in Monte Carlo type sampling and the designs of experiments. The paper studies estimation formulas for Pearson linear, Spearman and Kendall rank-order correlation coefficients and formulates the lower bounds on the performance of correlation control techniques such as the one presented in the companion paper Part I. In particular, probabilistic distributions of the two norms of correlation matrices defined in Part I are delivered for an arbitrary sample size and number of random variables in the case when the sampled values are ordered randomly. Next, an approximate number of designs with perfect uncorrelatedness is estimated based on the distribution of random correlation coefficients. It is shown that a large number of designs exist that perfectly match the unit correlation matrix.