Testing Conditional Independence via Rosenblatt Transforms

By Kyungchul Song. 2009

89 citations recorded by the source corpus.

This paper investigates the problem of testing conditional independence of Y and Z given λθ(X) for some unknown θ ∈ Θ ⊂ Rd, for a parametric function λθ(·). For instance, such a problem is relevant in recent literatures of heterogeneous treatment effects and contract theory. First, this paper finds that using Rosenblatt transforms in a certain way, we can construct a class of tests that are asymptotically pivotal and asymptotically unbiased against √n-converging Pitman local alternatives. The asymptotic pivotalness is convenient especially because the asymptotic critical values remain invariant over different estimators of the unknown parameter θ. Even when tests are asymptotically pivotal, however, it is often the case that simulation methods to obtain asymptotic critical values are yet unavailable or complicated, and hence this paper suggests a simple wild bootstrap procedure. A special case of the proposed testing framework is to test the presence of quantile treatment effects in a program evaluation data set. Using the JTPA training data set, we investigate the validity of nonexperimental procedures for inferences about quantile treatment effects of the job training program.

Subfields: Computer Networks and Communications, Finance, Statistics and Probability

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Citations in the corpus

Kernel-based Conditional Independence Test and Application in Causal Discovery (2011, methodology): “For instance, the method proposed by Song (2009) tests whether one can ﬁnd some (non-linear) function h and parameters θ 0 , such that X and Y are conditionally independent given a single index function λ θ 0 ( Z ) = h ( Z T θ 0 ) of Z .”
Testing Conditional Independence Via Empirical Likelihood (2014, unknown): “Song (2009) studies an asymptotically pivotal test of CI via the probability integral transform.”
Causal Effects of Monetary Shocks: Semiparametric Conditional Independence Tests with a Multinomial Propensity Score (2011, unknown): “Song (2009) considers a nonparametric test (as opposed to our semiparametric test) of conditional independence using the Rosenblatt transform.”
The conditional permutation test (2018, methodology): “The use of partial copulas was studied by Bergsma [3] and Song [18].”
On Azadkia–Chatterjee’s conditional dependence coefficient (2021, background): “…(2012) and Runge (2018) (on conditional mutual information); Székely and Rizzo (2014) and Wang et al. (2015) (on conditional distance correlation); Song (2009) and Cai et al. (2022) (based on Rosenblatt transformation); Bergsma (2004, 2011) and Veraverbeke et al. (2011) (copula-based); Hoyer et…”
Minimax optimal conditional independence testing (2020, methodology): “In a setting different from the present paper, Song [30] proposes a CI test for two variables given a single index of a random vector via “Rosenblatt transforms”, which are multivariate extensions of the probability integral transform.”
Testing functional inequalities (2011, background): “…approach (see, e.g., Khmaladze (1993), Stute, Thies and Zhu (1998), Bai (2003), and Khmaladze and Koul (2004)) or some tests of independence (or conditional independence) among random variables (see, e.g., Blum, Kiefer, and Rosenblatt (1961), Delgado and Mora (2000) and Song (2009)).”
Strongly consistent nonparametric tests of conditional independence (2012, methodology): “…testing conditional independence, in the context of index functions Linton and Gozalo (1996) proposed and investigated tests of Kolmogorov-Smirnov and Cramér-von Mises type with a generalization of empirical distribution functions using rectangles, while Song (2009) used Rosenblatt transforms.”