__ Summary and Contributions__: This paper proposes a test of quasi-independence (i.e. are X and Y independent, apart from the fact that, by design, X < Y), that applies in the right-censored setting. The new test consists in considering a generalization of a log rank test, where instead of having \omega fixed, the supremum over a rkhs ball is taken.

__ Strengths__: - the paper is clear and well written
- the mathematical aspects are correctly dealt with
- the applications targeted are clearly identified

__ Weaknesses__: - However, I found the overall contribution slightly insufficient for acceptance, as it focuses on a somewhat specific application (quasi independence under right-censoring)
- In particular, the differences between Sections 2 and 3 are not highlighted enough in my point of view, I almost had the feeling to read twice the same work
- Can authors motivate the choice of the factorising kernel, it seems a bit restrictive to decouple the effect on x and y?
- Can one imagine a kernel directly dealing with the censoring instead of considering the ^c functions?
- Censoring data can be seen as biased observations, coming from the original distribution biased through the multiplication by the indicator function 1{ Y < C}. Can more complex biasing functions be considered in the proposed analysis, and why? Possibly applying on x?
- The experiments on real data are not so stellar

__ Correctness__: Yes, I did not carefully check all mathematical proofs

__ Clarity__: Yes

__ Relation to Prior Work__: Yes, but the novelty seems a bit insufficient to me

__ Reproducibility__: Yes

__ Additional Feedback__: Response to rebuttal: I thank the authors for their feedback, that partially answered my concerns. I will not fight acceptance, and have updated my score to 5.

__ Summary and Contributions__: The quasi-independence (QI) model has been extensively studied, first in the contingency table literature and an extension to the continuous case, for truncated data, was given by Tsai (1990). This was further developed by Emura and Wang (2010). The present contribution builds forth on this work and develops a kernel test. An asymptotic analysis is given and a wild bootstrap is proposed.

__ Strengths__: The proposed kernel test seems to be a statistically natural and mathematically elegant way to test for QI. The approach is quite general and it seems to me supersedes previous work. The methodology is complete, giving both extensive theory and a testing algorithm.

__ Weaknesses__: None

__ Correctness__: Yes

__ Clarity__: Yes

__ Relation to Prior Work__: Yes

__ Reproducibility__: Yes

__ Additional Feedback__: It took me a while to understand exactly what was meant in the paper by QI. In the abstract, it says "...it is still of interest to determine whether there exists significant dependence beyond their ordering in time", a sentence I do not quite follow.
On first reading, I thought why not look at the dependence between X and Y-X. But the definition is given in (1), which if I understand correctly, means that if we have two (possibly hypothetical) independent variables X and Y, and only observe (X,Y) such that X<Y, then (1) follows. I felt some motivation for (1) can be given, a short derivation might be instructive. (I did get confused a bit initially that there was no conditioning on X<Y, but I understood after looking at Emura and Wang and the remark in line 98.)
Response to author feedback: I am satisfied with the response.

__ Summary and Contributions__: This paper considers observed data with (left-) truncation and (right-) censoring, and establishes kernel-based nonparametric tests of quasi-independence, i.e., the non-existence of dependence beyond the temporal ordering in which the variables are observed. The proposed approach is shown to recover existing parametric approaches as special cases, and experiments demonstrate the the test attains higher test power than existing approaches without sacrificing computational efficiency.

__ Strengths__: The paper addresses an interesting and practical problem, and proposes a nonparametric and theoretically justified approach. While there have been recent works on kernel-based two-sample and goodness-of-fit tests for censored data, the truncation setting studied in the current work poses additional challenges, and I believe the proposed quasi-independence tests could be of use to practitioners.

__ Weaknesses__: The proposed methodology is specifically designed for data with truncation and censoring, as existing approaches such as the HSIC have already addressed the more general scenarios. In practice, one major limitation of the proposed approach is that in such data with truncation and censoring, it may be likely that some form of even quasi-dependence would have crept into the data as e.g., caused by the underlying study design etc, which would (ideally) cause the test to reject the null-hypothesis. Thus, a more useful question might be to quantify the degree of quasi-independence and the extent to which it might affect downstream analyses, rather than a binary accept/reject decision. But of course, the paper presents an appealing first step in this direction.

__ Correctness__: Yes, to the best of my knowledge. I have skimmed through the supplementary material, but have not verified the proofs rigorously.

__ Clarity__: Yes, the paper is clearly written.

__ Relation to Prior Work__: Yes, the paper provides extensive discussion to existing parametric tests. I also find the review of the existing tests in the supplementary material helpful.

__ Reproducibility__: Yes

__ Additional Feedback__: EDIT: I thank the authors for their response, which have addressed my questions. I leave my evaluation unchanged.
=============================================
I have a few specific comments/questions:
- For equation (1), what are interesting/representative cases where other forms of F_X and S_Y are used beyond the common CDF and survival functions?
- In line 103, upon first reading, itâ€™s not immediately clear how the expression relates to log-rank test-statistics which are more commonly known in discrete cases. It might be helpful to briefly elaborate on the connection.
- In selecting/defining the kernel function, does the special care need to be taken to account for the underlying constrained spaced implied by the temporal ordering?
- While some runtime results are included in the supplementary material, it would be helpful to have a brief discussion/comparison of the theoretical computational complexities (big-O) of computing the test statistic for both the proposed and existing approaches.
- In experiments, how do WLR and WLR_SC fare in terms of computational cost?

__ Summary and Contributions__: The authors propose a nonparametric statistical test of quasi-independence which can be applied in the right-censored setting and has several real-life applications.
They also provide an asymptotic analysis of the test statistic and demonstrate in experiments that it obtains better power than existing approaches while being more computationally efficient.

__ Strengths__: EDIT: I thank the authors for their response, which have addressed my questions. I will keep the same score.
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There are several strengths to this paper.
1. The work is well-grounded in the theoretical perspective.
2. The empirical evaluation is strong where they have applied it challenging synthetic problems as well as real-world examples.
3. The method extends from the previous work [9] where they show a nonparametric generalization.
4. The method is highly relevant to the NeurIPS community due to its broad applications.

__ Weaknesses__: 1. My only concern is that this is a highly theoretical paper and most of the details are kept in the appendix. Due to the nature of the conference it might be hard for readers to go through such details. I would urge the authors to submit the full version to a journal.
2. The code does not open-sourced. It might be good if the authors could release the code and the datasets for ease of reproducibility.

__ Correctness__: The claims and empirical methodology seems to be correct, though I have not gone through the complete details in the Appendix

__ Clarity__: The paper is very well written and is very easy to follow.

__ Relation to Prior Work__: The paper contains a very good coverage of related literature and explicitly mention how they extend on it.

__ Reproducibility__: Yes

__ Additional Feedback__: