When scholars study joint distributions of multiple variables, copulas are useful. However, if the variables are not linearly correlated with each other yet are still not independent, most of conventional copulas are not up to the task. Examples include (inversed) U-shaped relationships and heteroskedasticity. To fill this gap, this manuscript sheds new light on a little-known copula, which I call the "normal mode copula." I characterize the copula's properties and show that the copula is asymmetric and nonmonotonic under certain conditions. I also apply the copula to a dataset about U.S. House vote share and campaign expenditure to demonstrate that the normal mode copula has better performance than other conventional copulas.

翻译：学者在研究多变量联合分布时，联结函数（Copula）是常用工具。然而，当变量间不存在线性相关却仍非独立时（例如倒U型关系及异方差性），多数传统联结函数无法胜任。为填补这一空白，本文重新审视一种鲜为人知的联结函数——我称之为"正态模式联结函数"（Normal Mode Copula）。本文刻画了该联结函数的性质，证明其在特定条件下呈现非对称性与非单调性。通过将其应用于美国众议院得票率与竞选开支数据集，实证表明正态模式联结函数相较其他传统联结函数具有更优性能。