Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

A central limit theorem for sums of functions of residuals in a high-dimensional regression model with an application to variance homoscedasticity test

  • Original Paper
  • Published:
TEST Aims and scope Submit manuscript

Abstract

We establish a joint central limit theorem for sums of squares and the fourth powers of residuals in a high-dimensional regression model. We then apply this CLT to detect the existence of heteroscedasticity for linear regression models without assuming randomness of covariates when the sample size n tends to infinity and the number of covariates p may be fixed or tend to infinity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Amari SV, Misra RB (1997) Closed-form expressions for distribution of sum of exponential random variables. IEEE Trans Reliab 46(4):519–522

    Article  Google Scholar 

  • Azzalini A, Bowman A (1993) On the use of nonparametric regression for checking linear relationships. J R Stat Soc Ser B Methodol 55(2):549–557

    MathSciNet  MATH  Google Scholar 

  • Bai Z, Silverstein JW (2010) Spectral analysis of large dimensional random matrices. Springer, Berlin

    Book  Google Scholar 

  • Breusch TS, Pagan AR (1979) A simple test for heteroscedasticity and random coefficient variation. Econom J Econom Soc 47(5):1287–1294

    MathSciNet  MATH  Google Scholar 

  • Cook RD, Weisberg S (1983) Diagnostics for heteroscedasticity in regression. Biometrika 70(1):1–10

    Article  MathSciNet  Google Scholar 

  • de Jong P (1987) A central limit theorem for generalized quadratic forms. Probab Theory Relat Fields 75(2):261–277

    Article  MathSciNet  Google Scholar 

  • Dette H, Munk A (1998) Testing heteroscedasticity in nonparametric regression. J R Stat Soc Ser B Stat Methodol 60(4):693–708

    Article  MathSciNet  Google Scholar 

  • Deya A, Nourdin I (2014) Invariance principles for homogeneous sums of free random variables. Bernoulli 20(2):586–603

    Article  MathSciNet  Google Scholar 

  • Glejser H (1969) A new test for heteroskedasticity. J Am Stat Assoc 64(325):316–323

    Article  Google Scholar 

  • Gotze F, Tikhomirov AN (1999) Asymptotic distribution of quadratic forms. Ann Probab 27(2):1072–1098

    Article  MathSciNet  Google Scholar 

  • Harrison MJ, McCabe BPM (1979) A test for heteroscedasticity based on ordinary least squares residuals. J Am Stat Assoc 74(366a):494–499

    Article  MathSciNet  Google Scholar 

  • Jensen DR, Solomon H (1972) A Gaussian approximation to the distribution of a definite quadratic form. J Am Stat Assoc 67(340):898–902

    MATH  Google Scholar 

  • John S (1971) Some optimal multivariate tests. Biometrika 58(1):123–127

    MathSciNet  MATH  Google Scholar 

  • Li Z, Yao J (2015) Homoscedasticity tests valid in both low and high-dimensional regressions. arXiv preprint arXiv:1510.00097

  • Liu H, Tang Y, Zhang HH (2009) A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Comput Stat Data Anal 53(4):853–856

    Article  MathSciNet  Google Scholar 

  • Nourdin I, Peccati G, Reinert G et al (2010) Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. Ann Probab 38(5):1947–1985

    Article  MathSciNet  Google Scholar 

  • Nourdin I, Peccati G, Poly G, Simone R (2016) Multidimensional limit theorems for homogeneous sums: a survey and a general transfer principle. ESAIM Probab Stat 20:293–308

    Article  MathSciNet  Google Scholar 

  • White H (1980) A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econom J Econom Soc 48(4):817–838

    MathSciNet  MATH  Google Scholar 

  • Whittle P (1964) On the convergence to normality of quadratic forms in independent variables. Theory Probab Appl 9(1):103–108

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yanqing Yin.

Additional information

Zhidong Bai is partially supported by a Grant NSF China 11571067 and 11471140. Guangming Pan was partially supported by a MOE Tier 2 Grant 2014-T2-2-060 and by a MOE Tier 1 Grant RG25/14 at the Nanyang Technological University, Singapore. Yanqing Yin was partially supported by a Grant NSF China 11701234, the Priority Academic Program Development of Jiangsu Higher Education Institutions and a project of China Scholarship Council.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bai, Z., Pan, G. & Yin, Y. A central limit theorem for sums of functions of residuals in a high-dimensional regression model with an application to variance homoscedasticity test. TEST 27, 896–920 (2018). https://doi.org/10.1007/s11749-017-0575-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-017-0575-x

Keywords

Mathematics Subject Classification