Abstract

This article treats the problem of constructing nonparametric asymp

totically distribution-free tests for conditional quantile independence in multidimensions. Our approach combines the quantile martingale difference divergence

with a notion of the recently introduced multivariate center-outward ranks and

signs. We derive the asymptotic null representation of the proposed test statistics by exploiting the degenerate V -type and U-type structures of the quantile

martingale difference divergence and the Glivenko-Cantelli strong consistency

and distribution-freeness of the center-outward ranks and signs. This representation permits direct calculation of limiting null distributions without requiring

bootstrap calibration. We further show that our center-outward versions of the

quantile martingale difference divergence tests are consistent against all fixed alternatives. A local power analysis provides strong support for the center-outward

approach by establishing the nontrivial power of our center-outward rank-based

tests over root-n neighborhoods. Moreover, the proposed tests are computationally feasible and well-defined without any moment assumptions. We illustrate

the advantages of the proposed methods via extensive simulation studies and a

gene expression dataset analysis.

Information

Preprint No.SS-2024-0266
Manuscript IDSS-2024-0266
Complete AuthorsKai Xu, Huijun Shi, Daojiang He
Corresponding AuthorsDaojiang He
Emailsdjhe@ahnu.edu.cn

References

  1. Baringhaus, L. and C. Franz (2004). On a new multivariate two-sample test. Journal of Multivariate Analysis 88, 190–206.
  2. Chernozhukov, V., A. Galichon, M. Hallin, and M. Henry (2017). Monge-kantorovich depth, quantiles, ranks and signs. The Annals of Statistics 45, 223–256.
  3. Chiang, A. P., J. S. Beck, H.-J. Yen, M. K. Tayeh, T. E. Scheetz, R. E. Swiderski, D. Y.
  4. Nishimura, T. A. Braun, K.-Y. A. Kim, J. Huang, et al. (2006). Homozygosity mapping with snp arrays identifies trim32, an e3 ubiquitin ligase, as a bardet–biedl syndrome gene (bbs11). Proceedings of the National Academy of Sciences 103, 6287–6292.
  5. Conde-Amboage, M., C. S´anchez-Sellero, and W. Gonz´alez-Manteiga (2015). A lack-of-fit test for quantile regression models with high-dimensional covariates. Computational Statistics and Data Analysis 88, 128–138.
  6. Dong, C., G. Li, and X. Feng (2019). Lack-of-fit tests for quantile regression models. Journal of the Royal Statistical Society Series B: Statistical Methodology 81, 629–648.
  7. Escanciano, J. C. and S. C. Goh (2014). Specification analysis of linear quantile models. Journal of Econometrics 178, 495–507.
  8. Escanciano, J. C. and C. Velasco (2010). Specification tests of parametric dynamic conditional quantiles. Journal of Econometrics 159, 209–221.
  9. Fan, J., Y. Feng, and R. Song (2011). Nonparametric independence screening in sparse ultrahigh-dimensional additive models. Journal of the American Statistical Association 106, 544–557.
  10. Hallin, M. (2017). On distribution and quantile functions, ranks and signs in rd: A measure transportation approach. Available at https://ideas.repec.org/p/eca/wpaper/2013258262.html.
  11. Hallin, M., E. del Barrio, J. Cuesta-Albertos, and C. Matr´an (2021). Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach. The Annals of Statistics 49, 1139–1165.
  12. He, X. and L. Zhu (2003). A lack-of-fit test for quantile regression. Journal of the American Statistical Association 98, 1013–1022.
  13. Horowitz, J. and V. G. Spokoiny (2002). An adaptive, rate-optimal test of linearity for median regression models. Journal of the American Statistical Association 97, 822–835.
  14. Koenker, R. (2005). Quantile Regression. Cambridge University Press, New York.
  15. Koenker, R. and G. Bassett (1978). Regression quantiles. Econometrica 46, 33–50.
  16. Lee, C. E. and H. Hilafu (2022). Quantile martingale difference divergence for dimension reduction. Statistica Sinica 32, 65–87.
  17. Lee, C. E. and X. Shao (2018). Martingale difference divergence matrix and its application to dimension reduction for stationary multivariate time series. Journal of the American Statistical Association 113, 216–229.
  18. Lee, C. E., X. Zhang, and X. Shao (2020). Testing the conditional mean independence for functional data. Biometrika 107, 331–346.
  19. Lehmann, E. L. and J. P. Romano (2005). Testing Statistical Hypotheses (third edition).
  20. Springer, New York.
  21. McKeague, I. and M. Qian (2015). An adaptive resampling test for detecting the presence of significant predictors. Journal of the American Statistical Association 110, 1422–1433.
  22. Otsu, T. (2008). Conditional empirical likelihood estimation and inference for quantile regression models. Journal of Econometrics 142, 508–538.
  23. Park, T., X. Shao, S. Yao, et al. (2015). Partial martingale difference correlation. Electronic Journal of Statistics 9, 1492–1517.
  24. Scheetz, T. E., K.-Y. A. Kim, R. E. Swiderski, A. R. Philp, T. A. Braun, K. L. Knudtson,
  25. A. M. Dorrance, G. F. DiBona, J. Huang, T. L. Casavant, et al. (2006). Regulation of gene expression in the mammalian eye and its relevance to eye disease. Proceedings of the National Academy of Sciences 103, 14429–14434.
  26. Serfling, R. (1980). Approximation Theorems of Mathematical Statistics. New York: Wiley.
  27. Shao, X. and J. Zhang (2014). Martingale difference correlation and its use in high-dimensional variable screening. Journal of the American Statistical Association 109, 1302–1318.
  28. Shi, H., M. Drton, and F. Han (2022). Distribution-free consistent independence tests via centeroutward ranks and signs. Journal of the American Statistical Association 117, 395–410.
  29. Shi, H., M. Hallin, M. Drton, and F. Han (2022). On universally consistent and fully distributionfree rank tests of vector independence. The Annals of Statistics 50, 1933–1959.
  30. Stute, W. (1997). Nonparametric model checks for regression. The Annals of Statistics 25, 613–641.
  31. Su, L. and X. Zheng (2017). A martingale-difference-divergence-based test for specification. Economics Letters 156, 162–167.
  32. Sz´ekely, G., M. Rizzo, and N. Bakirov (2007). Measuring and testing dependence by correlation of distances. The Annals of Statistics 35, 2769–2794.
  33. Sz´ekely, G. J. and M. L. Rizzo (2009). Brownian distance covariance. The Annals of Applied Statistics 3, 1236–1265.
  34. Sz´ekely, G. J. and M. L. Rizzo (2014). Partial distance correlation with methods for dissimilarities. The Annals of Statistics 42, 2382–2412.
  35. van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press, Cambridge.
  36. Wang, H., I. McKeague, and M. Qian (2018). Testing for marginal linear effects in quantile regression. Journal of the Royal Statistical Society: Series B 80, 433–452.
  37. Wang, L., Y. Wu, and R. Li (2012). Quantile regression for analyzing heterogeneity in ultra-high dimension. Journal of the American Statistical Association 107, 214–222.
  38. Whang, Y.-J. (2006). Smoothed empirical likelihood methods for quantile regression models. Econometric Theory 22, 173–205.
  39. Xu, K. and N. An (2024). A tuning-free efficient test for marginal linear effects in highdimensional quantile regression. Annals of the Institute of Statistical Mathematics 76, 93–110.
  40. Xu, K. and F. Chen (2020). Martingale-difference-divergence-based tests for goodness-of-fit in quantile models. Journal of Statistical Planning and Inference 207, 138154.
  41. Yao, S., X. Zhang, X. Shao, et al. (2018). Testing mutual independence in high dimension via distance covariance. Journal of the Royal Statistical Society Series B 80, 455–480.
  42. Zhang, X., S. Yao, and X. Shao (2018). Conditional mean and quantile dependence testing in high dimension. The Annals of Statistics 46, 219–246.
  43. Zheng, J. (1998). A consistent nonparametric test of parametric regression models under conditional quantile restrictions. Econometric Theory 14, 123–138.
  44. Zhou, Y., K. Xu, L. Zhu, and R. Li (2024). Rank-based indices for testing independence between two high-dimensional vectors. The Annals of Statistics 52, 184–206. Kai Xu

Acknowledgments

Kai Xu is supported by National Natural Science Foundation of China

(12271005, 11901006), Natural Science Foundation of Anhui Province (2308085Y06,

1908085QA06) and Young Scholars Program of Anhui Province (2023).

by National Natural Science Foundation of China (11201005) and Natural

Science Foundation of Anhui Province (2408085MA005).

Supplementary Materials

The online supplementary material contains all technical proofs, and more

numerical results on some aspects of limiting distributions and comparison

under moderate dimension.

Supplementary materials are available for download.