Inference for Partially Linear Quantile Regression Models in Ultrahigh Dimension

Abstract

Conditional quantile regression provides a useful statistical tool for modeling and inferring the relationship between the response and covariates in the heterogeneous data. In this paper, we develop a novel testing procedure for the ultrahigh-dimensional partially linear quantile regression model to investigate the significance of ultrahigh-dimensional interested covariates in the presence of ultrahigh-dimensional nuisance covariates. The proposed test statistic is an $L_2$-type statistic. We estimate the nonparametric component by some flexible machine learners to handle the complexity and ultrahigh dimensionality of considered models. We establish the asymptotic normality of the proposed test statistic under the null and local alternative hypotheses. A screening-based testing procedure is further provided to make our test more powerful in practice under the ultrahigh-dimensional regime. We evaluate the finite-sample performance of the proposed method via extensive simulation studies. A real application to a breast cancer dataset is presented to illustrate the proposed method.

Keywords

Semiparametric model / Significance testing / Quantile regression / Ultrahigh dimensionality

References

[1] Belloni, A., Chernozhukov, V.: $l_1$-penalized quantile regression in high-dimensional sparse models. Ann. Stat. 39(1), 82–130 (2011)

[2] Beyerlein A, Kries R, Ness AR, Ong KK. Genetic markers of obesity risk: stronger associations with body composition in overweight compared to normal-weight children. PLoS ONE. 2011, 6 4 19057

[3] Cai L, Guo X, Li G, Tan F. Tests for high-dimensional single-index models. Electron. J. Stat.. 2023, 17 1 429-463

[4] Chen SX, Qin Y-L. A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Stat.. 2010, 38 2 808-835

[5] Chen J, Li Q, Chen HY. Testing generalized linear models with high-dimensional nuisance parameters. Biometrika. 2023, 110 1 83-99

[6] Chernozhukov V, Chetverikov D, Demirer M, Duflo E, Hansen C, Newey W, Robins J. Double/debiased machine learning for treatment and structural parameters. Econometr. J.. 2018, 21 1 1-68

[7] Cui H, Guo W, Zhong W. Test for high-dimensional regression coefficients using refitted cross-validation variance estimation. Ann. Stat.. 2018, 46 3 958-988

[8] Cui, H., Zou, F., Ling, L.: Feature screening and error variance estimation for ultrahigh-dimensional linear model with measurement errors. Commun. Math. Stat., pp. 1–33 (2023)

[9] Dezeure R, Bühlmann P, Zhang C-H. High-dimensional simultaneous inference with the bootstrap. TEST. 2017, 26 685-719

[10] Du L, Guo X, Sun W, Zou C. False discovery rate control under general dependence by symmetrized data aggregation. J. Am. Stat. Assoc.. 2023, 118 541 607-621

[11] Fan J, Lv J. Sure independence screening for ultrahigh dimensional feature space. J. R. Stat. Soc. Ser. B. 2008, 70 5 849-911

[12] Fan J, Guo S, Hao N. Variance estimation using refitted cross-validation in ultrahigh dimensional regression. J. R. Stat. Soc. Ser. B. 2012, 74 1 37-65

[13] Guo B, Chen SX. Tests for high dimensional generalized linear models. J. R. Stat. Soc. Ser. B. 2016, 78 5 1079-1102

[14] Guo W, Zhong W, Duan S, Cui H. Conditional test for ultrahigh dimensional linear regression coefficients. Stat. Sin.. 2022, 32 1381-1409

[15] Hall P, Heyde CC. Martingale Limit Theory and Its Application. 2014 UK: Academic Press

[16] Khaled W, Lin J, Han Z, Zhao Y, Hao H. Test for heteroscedasticity in partially linear regression models. J. Syst. Sci. Complex.. 2019, 32 1194-1210

[17] Koenker R. Quantile Regression. 2005 Cambridge: Cambridge University Press

[18] Koenker R, Bassett G. Regression quantiles. Econometrica. 1978, 46 1 33-50

[19] Koenker R, Chernozhukov V, He X, Peng L. Handbook of Quantile Regression. 2017 New York: CRC Press

[20] Lu W, Zhu Z, Lian H. Sparse and low-rank matrix quantile estimation with application to quadratic regression. Stat. Sin.. 2023, 33 2 945-959

[21] Ma R, Cai T, Li H. Global and simultaneous hypothesis testing for high-dimensional logistic regression models. J. Am. Stat. Assoc.. 2021, 116 534 984-998

[22] Meinshausen N, Meier L, Bühlmann P. P-values for high-dimensional regression. J. Am. Stat. Assoc.. 2009, 104 488 1671-1681

[23] Méndez Civieta, Á., Aguilera-Morillo, M.C., Lillo, R.E.: Asgl: a python package for penalized linear and quantile regression. arXiv preprint arXiv:2111.00472 (2021)

[24] Ning Y, Liu H. A general theory of hypothesis tests and confidence regions for sparse high dimensional models. Ann. Stat.. 2017, 45 1 158-195

[25] Parker JS, Mullins M, Cheang MC, Leung S, Voduc D, Vickery T, Davies S, Fauron C, He X, Hu Z. Supervised risk predictor of breast cancer based on intrinsic subtypes. J. Clin. Oncol.. 2009, 27 8 1160-1167

[26] Prat A, Bianchini G, Thomas M, Belousov A, Cheang MC, Koehler A, Gómez P, Semiglazov V, Eiermann W, Tjulandin S. Research-based PAM50 subtype predictor identifies higher responses and improved survival outcomes in HER2-positive breast cancer in the NOAH study. Clin. Cancer Res.. 2014, 20 2 511-521

[27] Sherwood B, Wang L. Partially linear additive quantile regression in ultra-high dimension. Ann. Stat.. 2016, 44 1 288-317

[28] Shi, H., Sun, B., Yang, W., Guo, X.: Tests for ultrahigh-dimensional partially linear regression models. arXiv preprint arXiv:2304.07546 (2023)

[29] Song X, Li G, Zhou Z, Wang X, Ionita-Laza I, Wei Y. QRank: a novel quantile regression tool for eQTL discovery. Bioinformatics. 2017, 33 14 2123-2130

[30] Tan F, Jiang X, Guo X, Zhu L. Testing heteroscedasticity for regression models based on projections. Stat. Sin.. 2021, 31 2 625-646

[31] Tang Y, Wang Y, Judy Wang H, Pan Q. Conditional marginal test for high dimensional quantile regression. Stat. Sin.. 2022, 32 869-892

[32] Wang HJ, Zhu Z, Zhou J. Quantile regression in partially linear varying coefficient models. Ann. Stat.. 2009, 37 6B 3841-3866

[33] Wang, H., Jin, H., Jiang, X.: Feature selection for high-dimensional varying coefficient models via ordinary least squares projection. Commun. Math. Stat., pp. 1–42 (2023)

[34] Wu Y, Yin G. Conditional quantile screening in ultrahigh-dimensional heterogeneous data. Biometrika. 2015, 102 1 65-76

[35] Yang, W., Guo, X., Zhu, L.: Score function-based tests for ultrahigh-dimensional linear models. arXiv preprint arXiv:2212.08446 (2022)

[36] Zhang X, Cheng G. Simultaneous inference for high-dimensional linear models. J. Am. Stat. Assoc.. 2017, 112 518 757-768

[37] Zhang Y, Lian H, Yu Y. Ultra-high dimensional single-index quantile regression. J. Mach. Learn. Res.. 2020, 21 1 9212-9236

[38] Zhong P-S, Chen SX. Tests for high-dimensional regression coefficients with factorial designs. J. Am. Stat. Assoc.. 2011, 106 493 260-274

Funding

National Natural Science Foundation of China(12071038)