A New Test for Large Dimensional Regression Coefficients

ABSTRACT

In the article, hypothesis test for coefficients in high dimensional regression models is considered. I develop simultaneous test statistic for the hypothesis test in both linear and partial linear models. The derived test is designed for growing p and fixed n where the conventional*F*-test is no longer appropriate. The asymptotic distribution of the proposed test statistic under the null hypothesis is obtained.

In the article, hypothesis test for coefficients in high dimensional regression models is considered. I develop simultaneous test statistic for the hypothesis test in both linear and partial linear models. The derived test is designed for growing p and fixed n where the conventional

Cite this paper

nullJ. Luo and Y. Zuo, "A New Test for Large Dimensional Regression Coefficients,"*Open Journal of Statistics*, Vol. 1 No. 3, 2011, pp. 212-216. doi: 10.4236/ojs.2011.13025.

nullJ. Luo and Y. Zuo, "A New Test for Large Dimensional Regression Coefficients,"

References

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[3] S. Chen and Y. Qin, “A Two Sample Test for High Di- mensional Data with Applications to Gene-set Testing,” Annals of Statistics, Vol. 38, No. 2, 2010, pp. 808-835. doi:10.1214/09-AOS716

[4] P. Zhong and S. Chen, “Tests for High-Dimensional Re- gression Coefficients with Factorial Designs,” Journal of American Statistical Association, Vol. 106, No. 493, 2011, pp. 260-274. doi:10.1198/jasa.2011.tm10284

[5] R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Ser. B, Vol. 58, No. 1, 1996, pp. 267-288.

[6] J. Fan and J. Lv, “Sure Independence Screening for Ul- tra-high Dimensional Feature Space (with Discussion),” Journal of Royal Statistical Society, Vol. 70, No. 5, 2008, pp. 849-911. doi:10.1111/j.1467-9868.2008.00674.x

[7] J. Shao and S. Chow, “Variable Screening in Predicting Clinical Outcome with High-Dimensional Microarrays,” Journal of Multivariate Analysis, Vol. 98, No. 8, 2007, pp. 1529-1538. doi:10.1016/j.jmva.2004.12.004

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[9] T. Severini and W. Wong, “Profile Likelihood and Con- ditionally Parametric Models,” Annals of Statistics, Vol. 20, No. 4, 1992, pp. 1768-1802. doi:10.1214/aos/1176348889

[10] J. Rice, “Bandwidth Choice for Nonparametric Regres- sion,” Annals of Statistics, Vol. 12, No. 4, 1984, pp. 1215- 1230. doi:10.1214/aos/1176346788

[11] J. Horowitz and V. Spokoiny, “An Adaptive Rate-optimal Test of a Parametric Mean-regression Model against a Nonparametric Alternative,” Econometrica, Vol. 69, No. 3, 2001, pp. 599-631. doi:10.1111/1468-0262.00207

[12] C. Rao, H. Touteburg, and C. Heumann, “Linear Models and Generalizations,” Springer, New York, 2008.

[13] A. Hoerl and R. Kennard, “Ridge Regression Biased Estimation for Nonorthogonal Problems,” Technometrics, Vol. 12, No. 1, 1970, pp. 55-67. doi:10.2307/1267351

[14] J. Luo, “The Discovery of Mean Square Error Consis- tency of Ridge Estimator,” Statistics and Probability Let- ters, Vol. 80, No. 5, 2010, pp. 343-347. doi:10.1016/j.spl.2009.11.008

[15] J. Luo, “Asymptotical Properties of Coefficient of De- termination for Ridge Regression with Growing Dimen- sions,” Oriental Journal of Statistical Methods, Theory and Applications, Vol. 1, No. 1, 2011, pp. 41-49.

[16] L. Wang, L. Brown and T. Cai, “A Difference Based Approach to Semiparametric Partial Linear Model,” Elec- tronic Journal of Statistics, Vol. 5, 2011, pp. 619-641.

[17] A. Yatchew, “An Elementary Estimator of the Partial Linear Model,” Economics Letters, Vol. 57, No. 2, 1997, pp. 135-143. doi:10.1016/S0165-1765(97)00218-8

[18] A. Yatchew, “Scale Economies in Electricity Distribution: A Semiparametric Analysis,” Journal of Applied Eco- nomics, Vol. 15, No. 2, 2000, pp. 187-210.

[19] J. Luo, “Asymptotic Efficiency of Ridge Estimator in Linear and Semiparametric Linear Models,” Statistics and Probability Letters, Vol. 82, No. 1, 2011, pp.58-62. doi:10.1016/j.spl.2011.08.018.

[1] M. Kosorok and S. Ma, “Marginal Asymptotics for the ‘Large p, Small n’ Paradigm: With Aplications to Mi- croarray Data,” Annals of Statistics, Vol. 35, No. 4, 2007, pp. 1456-1486. doi:10.1214/009053606000001433

[2] J. Fan, P. Hall and Q. Yao, “To How Many Simultaneous Hypothesis Tests Can Normal Student’s t or Bootstrap Calibrations Be Applied,” Journal of the American Sta- tistical Association, Vol. 102, No. 480, 2007, pp. 1282- 1288. doi:10.1198/016214507000000969

[3] S. Chen and Y. Qin, “A Two Sample Test for High Di- mensional Data with Applications to Gene-set Testing,” Annals of Statistics, Vol. 38, No. 2, 2010, pp. 808-835. doi:10.1214/09-AOS716

[4] P. Zhong and S. Chen, “Tests for High-Dimensional Re- gression Coefficients with Factorial Designs,” Journal of American Statistical Association, Vol. 106, No. 493, 2011, pp. 260-274. doi:10.1198/jasa.2011.tm10284

[5] R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Ser. B, Vol. 58, No. 1, 1996, pp. 267-288.

[6] J. Fan and J. Lv, “Sure Independence Screening for Ul- tra-high Dimensional Feature Space (with Discussion),” Journal of Royal Statistical Society, Vol. 70, No. 5, 2008, pp. 849-911. doi:10.1111/j.1467-9868.2008.00674.x

[7] J. Shao and S. Chow, “Variable Screening in Predicting Clinical Outcome with High-Dimensional Microarrays,” Journal of Multivariate Analysis, Vol. 98, No. 8, 2007, pp. 1529-1538. doi:10.1016/j.jmva.2004.12.004

[8] R. Carroll, J. Fan, I. Gijbels and M. Wand, “Generalized Partially Linear Single-Index Models,” Journal of Ameri- can Statistical Association, Vol. 92, No. 438, 1997, pp. 477-489. doi:10.2307/2965697

[9] T. Severini and W. Wong, “Profile Likelihood and Con- ditionally Parametric Models,” Annals of Statistics, Vol. 20, No. 4, 1992, pp. 1768-1802. doi:10.1214/aos/1176348889

[10] J. Rice, “Bandwidth Choice for Nonparametric Regres- sion,” Annals of Statistics, Vol. 12, No. 4, 1984, pp. 1215- 1230. doi:10.1214/aos/1176346788

[11] J. Horowitz and V. Spokoiny, “An Adaptive Rate-optimal Test of a Parametric Mean-regression Model against a Nonparametric Alternative,” Econometrica, Vol. 69, No. 3, 2001, pp. 599-631. doi:10.1111/1468-0262.00207

[12] C. Rao, H. Touteburg, and C. Heumann, “Linear Models and Generalizations,” Springer, New York, 2008.

[13] A. Hoerl and R. Kennard, “Ridge Regression Biased Estimation for Nonorthogonal Problems,” Technometrics, Vol. 12, No. 1, 1970, pp. 55-67. doi:10.2307/1267351

[14] J. Luo, “The Discovery of Mean Square Error Consis- tency of Ridge Estimator,” Statistics and Probability Let- ters, Vol. 80, No. 5, 2010, pp. 343-347. doi:10.1016/j.spl.2009.11.008

[15] J. Luo, “Asymptotical Properties of Coefficient of De- termination for Ridge Regression with Growing Dimen- sions,” Oriental Journal of Statistical Methods, Theory and Applications, Vol. 1, No. 1, 2011, pp. 41-49.

[16] L. Wang, L. Brown and T. Cai, “A Difference Based Approach to Semiparametric Partial Linear Model,” Elec- tronic Journal of Statistics, Vol. 5, 2011, pp. 619-641.

[17] A. Yatchew, “An Elementary Estimator of the Partial Linear Model,” Economics Letters, Vol. 57, No. 2, 1997, pp. 135-143. doi:10.1016/S0165-1765(97)00218-8

[18] A. Yatchew, “Scale Economies in Electricity Distribution: A Semiparametric Analysis,” Journal of Applied Eco- nomics, Vol. 15, No. 2, 2000, pp. 187-210.

[19] J. Luo, “Asymptotic Efficiency of Ridge Estimator in Linear and Semiparametric Linear Models,” Statistics and Probability Letters, Vol. 82, No. 1, 2011, pp.58-62. doi:10.1016/j.spl.2011.08.018.