Effects of statistical distribution on nonlinear correlation coefficient

Nonlinear Correlation Coefficient (NCC) is proposed to quantitatively measure the nonlinear relation between the variables. Much attention paid on NCC leads to application being far from the basic research. In this paper, we propose a mathematical framework to systematically analyze the effects of statistical distribution on NCC. It is found that the arbitrary distortion on distribution leads to the decreasing of NCC and the sharing structure of the concerned distribution minimizes the NCC. Furthermore, considering the situation of a sequence of discrete observations, we provide a collection of rigorous logistic results to justify NCC is an increasing function of sequence length. The numerical examples on Lorenz and linear auto-regression system are used to illustrate proposed results suitably and effectively.