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Metrika

, Volume 70, Issue 1, pp 59–77 | Cite as

Heteroscedasticity diagnostics for t linear regression models

  • Jin-Guan LinEmail author
  • Li-Xing Zhu
  • Feng-Chang Xie
Article

Abstract

The t regression models provide a useful extension of the normal regression models for datasets involving errors with longer-than-normal tails. Homogeneity of variances (if they exist) is a standard assumption in t regression models. However, this assumption is not necessarily appropriate. This paper is devoted to tests for heteroscedasticity in general t linear regression models. The asymptotic properties, including asymptotic Chi-square and approximate powers under local alternatives of the score tests, are studied. Based on the modified profile likelihood (Cox and Reid in J R Stat Soc Ser B 49(1):1–39, 1987), an adjusted score test for heteroscedasticity is developed. The properties of the score test and its adjustment are investigated through Monte Carlo simulations. The test methods are illustrated with land rent data (Weisberg in Applied linear regression. Wiley, New York, 1985).

Keywords

Adjusted score test Approximate local powers Asymptotic properties Heteroscedasticity Score test t Regression models Simulation studies 

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References

  1. Aitkin M (1987). Modelling variance heterogeneity in normal regression using GLIM. Appl Stat 36: 332–339 CrossRefGoogle Scholar
  2. Barroso LP and Cordeiro GM (2005). Bartlett corrections in heteroskedastic t regression models. Stat Probab Lett 75: 86–96 zbMATHCrossRefMathSciNetGoogle Scholar
  3. Chen CF (1983). Score test for regression models. J Am Stat Assoc 78: 158–161 zbMATHCrossRefGoogle Scholar
  4. Cook RD and Weisberg S (1983). Diagnostics for heteroscedasticity in regression. Biometrika 70: 1–10 zbMATHCrossRefMathSciNetGoogle Scholar
  5. Cox DR and Hinkley DV (1974). Theoretical statistics. Chapman and Hall, London zbMATHGoogle Scholar
  6. Cox DR and Reid N (1987). Parameter orthogonality and approximate conditional inference. J R Stat Soc Ser B 49(1): 1–39 zbMATHMathSciNetGoogle Scholar
  7. Cysneiros FJA, Paula GA and Galea M (2007). Heteroscedastic symmetrical linear models. Stat Probab Lett 77: 1084–1090 zbMATHCrossRefMathSciNetGoogle Scholar
  8. Diblasi A and Bowman A (1997). Testing for constant variance in a linear model. Stat Probab Lett 39: 95–103 CrossRefGoogle Scholar
  9. Eubank RL and Thomas W (1993). Detecting heteroscedasticity in nonparametric regression. J R Stat Soc Ser B 55: 145–155 zbMATHMathSciNetGoogle Scholar
  10. Hutton JL and Solomon PJ (1997). Parameter orthogonality in mixed regression models for survival data. J R Stat Soc Ser B 59(1): 125–136 zbMATHCrossRefMathSciNetGoogle Scholar
  11. Lange KL, Little RJA and Taylor JMG (1989). Robust statistical modelling using the t distribution. J Am Stat Assoc 84: 881–896 CrossRefMathSciNetGoogle Scholar
  12. Lin JG and Wei BC (2003). Testing for heteroscedasticity in nonlinear regression model. Commun Stat Theory Methods 32(1): 171–192 CrossRefMathSciNetGoogle Scholar
  13. Lin JG and Wei BC (2004). Testing for heteroscedasticity and correlation in nonlinear regression with correlated errors. Commun Stat Theory Methods 33(2): 251–275 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Lin JG and Wei BC (2006). Approximate power of score test for variance heterogeneity under local alternatives in nonlinear models. Comput Stat Data Anal 50: 3179–3198 zbMATHCrossRefMathSciNetGoogle Scholar
  15. Lin JG, Wei BC and Zhang NS (2004). Varying dispersion diagnostics for inverse gaussian regression models. J Appl Stat 31: 1157–1170 zbMATHCrossRefMathSciNetGoogle Scholar
  16. Liu CH and Rubin DB (1995). ML estimation of the t distribution using EM and its extensions, ECM and ECME. Stat Sin 5: 19–39 zbMATHMathSciNetGoogle Scholar
  17. Lubin JH and Gail MH (1990). On power and sample size for studying features of the relative odds of disease. Am J Epidemiol 131: 552–566 Google Scholar
  18. Parker RA and Bregman DJ (1986). Sample size for individually matched case-control studies. Biometrics 42: 919–926 zbMATHCrossRefGoogle Scholar
  19. Self SG and Mauritsen R (1988). Power/sample size calculations for generalized linear models. Biometrics 44: 79–86 zbMATHCrossRefMathSciNetGoogle Scholar
  20. Self SG, Mauritsen R and Ohara J (1992). Power calculations for likelihood ratio tests in generalized linear models. Biometrics 48: 31–39 CrossRefGoogle Scholar
  21. Shoham S (2002). Robust clustering by eterministic agglomeration EM of mixtures of multivariate t distributions. Pattern Recognit 35: 1127–1142 zbMATHCrossRefGoogle Scholar
  22. Simonoff JS and Tsai CL (1994). Improved tests for nonconstant variance in regression based on the modified profile likelihood. Appl Stat 43: 357–370 zbMATHCrossRefMathSciNetGoogle Scholar
  23. Smyth GK (1989). Generalized linear models with varying dispersion. J R Stat Soc Ser B 51: 47–60 MathSciNetGoogle Scholar
  24. Tsai CL (1986). Score test for the first-order autoregressive model with heteroscedasticity. Biometrika 73: 455–460 CrossRefMathSciNetGoogle Scholar
  25. Taylor JT and Verbyla AP (2004). Joint modelling of location and scale parameters of the t distribution. Stat Modell 4: 91–112 zbMATHCrossRefMathSciNetGoogle Scholar
  26. Verbyla AP (1993). Modelling variance heterogeneity:residual maximum likelihood and diagnostics. J R Stat Soc Ser B 55: 509–521 MathSciNetGoogle Scholar
  27. Weisberg S (1985). Applied linear regression. Wiley, New York zbMATHGoogle Scholar
  28. Wei BC, Shi JQ, Fung WK and Hu YQ (1998). Testing for varying dispersion in exponential family nonlinear models. Ann Inst Stat Math 50: 277–294 zbMATHCrossRefMathSciNetGoogle Scholar
  29. Woldie M, Folks JL and Chandler JP (2001). Power function for inverse Gaussian regression models. Commun Stat Theory Methods 30(5): 787–797 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsSoutheast UniversityNanjingChina
  2. 2.Department of MathematicsHong Kong Baptist UniversityJiulongChina
  3. 3.Department of MathematicsNanjing Agricultural UniversityNanjingChina

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