Copulas are helpful in studying joint distributions of two variables, in particular, whenconfounders are unobserved. However, most conventional copulas cannot model joint distributions where one variable does not increase or decrease in the other in a monotonic manner. For instance, suppose that two variables are linearly positively correlated for one type of unit and negatively for another type of unit. If the type is unobserved, we can observe only a mixture of both types. Seemingly, one variable tends to take either a high or low value (or a middle value) when the other variable is small (large), or vice versa. To address this issue, I consider an overlooked copula with trigonometric functions (Chesneau [2021, Applied Mathematics, 1(1), pp. 3–17]) thatI name the “normal mode copula.” I apply the copula to a dataset about government formation and duration to demonstrate that the normal mode copula has better performance than other conventional copulas.

Course keeping for ships is the core of automatic navigation in sea transportation. Much work has concentrated on developing novel control strategies for closed loop systems. We have turned our attention the other way to improve the control performance of marine autopilots in this work by “modulating” the control error using a sine function while the construction of the controller is not changed. The nonlinear feedback signal thus obtained is sent to the controller to replace the control error itself, which used to be the deviation between the output response and the reference input of the system. Such a control scheme is called “nonlinear feedback control” hereafter. Theoretical analysis by using a describing function and robust control theory shows that the same control quality is guaranteed with minor control actions for the nonlinear feedback scheme. Simulation experiments were carried out for the ship Yulong of Dalian Maritime University. It is shown that the method postulated in this paper has advantages of safety and energy saving in navigation; the maximum initial rudder angle is reduced by 31·2% with satisfactory control effect.