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arXiv · 2512.19641

Testing for Conditional Independence in Binary Single-Index Models

John H. J. Einmahl·Denis Kojevnikov·Bas J. M. Werker
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Abstract

We wish to test whether a real-valued variable $Z$ has explanatory power, in addition to a multivariate variable $X$, for a binary variable $Y$. Thus, we are interested in testing the hypothesis $\mathbb{P}(Y=1\, | \, X,Z)=\mathbb{P}(Y=1\, | \, X)$, based on $n$ i.i.d.\ copies of $(X,Y,Z)$. In order to avoid the curse of dimensionality, we follow the common approach of assuming that the dependence of both $Y$ and $Z$ on $X$ is through a single-index $X^\top\beta$ only. Splitting the sample on both $Y$-values, we construct a two-sample empirical process of transformed $Z$-variables, after splitting the $X$-space into parallel strips. Studying this two-sample empirical process is challenging: it does not converge weakly to a standard Brownian bridge, but after an appropriate normalization it does. We use this result to construct distribution-free tests.

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John H. J. Einmahl, Denis Kojevnikov, Bas J. M. Werker. 2025-12-22. Testing for Conditional Independence in Binary Single-Index Models. https://arxiv.org/abs/2512.19641

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