The cross ratio function (CRF) is a commonly used tool to describe local dependence between two correlated variables. Being a ratio of conditional hazards, the CRF can be rewritten in terms of (first and second derivatives of) the survival copula of these variables. Bernstein estimators for (the derivatives of) this survival copula are used to define a nonparametric estimator of the cross ratio, and asymptotic normality thereof is established. We consider simulations to study the finite sample performance of our estimator for copulas with different types of local dependency. A real dataset is used to investigate the dependence between food expenditure and net income. The estimated CRF reveals that families with a low net income relative to the mean net income will spend less money to buy food compared to families with larger net incomes. This dependence, however, disappears when the net income is large compared to the mean income.

中文翻译：

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交叉比函数的非参数估计

交叉比函数 (CRF) 是一种常用的工具，用于描述两个相关变量之间的局部依赖性。作为条件风险的比率，CRF 可以根据这些变量的生存联结（一阶和二阶导数）重写。该生存联结（的导数）的 Bernstein 估计量用于定义交叉比的非参数估计量，并建立其渐近正态性。我们考虑模拟来研究我们的估计器对具有不同类型局部依赖性的 copula 的有限样本性能。一个真实的数据集用于调查食品支出和净收入之间的依赖关系。估计的 CRF 显示，与净收入较高的家庭相比，净收入相对于平均净收入较低的家庭购买食物的钱较少。