Annals of the Institute of Statistical Mathematics

, Volume 70, Issue 5, pp 1147–1175 | Cite as

A generalized partially linear framework for variance functions

  • Yixin Fang
  • Heng Lian
  • Hua Liang


When model the heteroscedasticity in a broad class of partially linear models, we allow the variance function to be a partial linear model as well and the parameters in the variance function to be different from those in the mean function. We develop a two-step estimation procedure, where in the first step some initial estimates of the parameters in both the mean and variance functions are obtained and then in the second step the estimates are updated using the weights calculated based on the initial estimates. The resulting weighted estimators of the linear coefficients in both the mean and variance functions are shown to be asymptotically normal, more efficient than the initial un-weighted estimators, and most efficient in the sense of semiparametric efficiency for some special cases. Simulation experiments are conducted to examine the numerical performance of the proposed procedure, which is also applied to data from an air pollution study in Mexico City.


Efficiency Generalized least squares Generalized partially linear model Kernel regression Profiling Variance function 


  1. Bickel, P. (1978). Using residuals robustly: Tests for heteroscedasticity, nonlinearity. The Annals of Statistics, 6, 266–291.MathSciNetCrossRefMATHGoogle Scholar
  2. Bickel, P. J., Klaasen, C. A. J., Ritov, Y., Wellner, J. A. (1993). Efficient and adaptive estimation for semiparametric models. Baltimore: The Johns Hopkins University Press.Google Scholar
  3. Box, G., Hill, W. (1974). Correcting inhomogeneity of variance with power transformation weighting. Technometrics, 16, 385–389.Google Scholar
  4. Box, G., Meyer, D. (1986). An analysis for unreplicated fractional factorials. Technometrics, 28, 11–18.Google Scholar
  5. Cai, T. T., Wang, L. (2008). Adaptive variance function estimation in heteroscedastic nonparametric regression. Annals of Statistics, 36(5), 2025–2054.Google Scholar
  6. Carroll, R. J. (1982). Adapting for heteroscedasticity in linear models. The Annals of Statistics, 10, 1224–1233.MathSciNetCrossRefMATHGoogle Scholar
  7. Carroll, R. J. (2003). Variances are not always nuisance parameters. Biometrics, 59(2), 211–220.MathSciNetCrossRefMATHGoogle Scholar
  8. Carroll, R. J., Fan, J., Gijbels, I., Wand, M. P. (1997). Generalized partially linear single-index models. Journal of the American Statistical Association, 92(438), 477–489.Google Scholar
  9. Carroll, R. J., Härdle, W. (1989). Second order effects in semiparametric weighted least squares regression. Statistics, 2, 179–186.Google Scholar
  10. Carroll, R. J., Ruppert, D. (1982). Robust estimation in heteroscedasticity linear models. The Annals of Statistics, 10, 429–441.Google Scholar
  11. Carroll, R., Ruppert, D. (1988). Transformation and weighting in regression. New York: Chapman & Hall.Google Scholar
  12. Chen, H. (1988). Convergence rates for parametric components in a partly linear model. The Annals of Statistics, 16, 136–146.MathSciNetCrossRefMATHGoogle Scholar
  13. Davidian, M., Carroll, R. J. (1987). Variance function estimation. Journal of the American Statistical Association, 82, 1079–1091.Google Scholar
  14. Engle, R. F., Granger, C. W. J., Rice, J., Weiss, A. (1986). Semiparametric estimates of the relation between weather and electricity sales. Journal of the American Statistical Association, 81(394), 310–320.Google Scholar
  15. Fuller, W., Rao, J. (1978). Estimation for a linear regression model with unknown diagonal covariance matrix. The Annals of Statistics, 6, 1149–1158.Google Scholar
  16. Hall, P., Carroll, R. J. (1989). Variance function estimation in regression—The effect of estimating the mean. Journal of the Royal Statistical Society Series B-Methodological, 51(1), 3–14.Google Scholar
  17. Härdle, W., Mammen, E., Müller, M. (1998). Testing parametric versus semiparametric modeling in generalized linear models. Journal of the American Statistical Association, 93, 1461–1474.Google Scholar
  18. Härdle, W., Liang, H., Gao, J. (2000). Partially linear models. Heidelberg: Physica-Verlag.Google Scholar
  19. Härdle, W., Müller, M., Sperlich, S., Werwatz, A. (2004). Nonparametric and semiparametric models. New York: Springer.Google Scholar
  20. Heckman, N. E. (1986). Spline smoothing in partly linear models. Journal of the Royal Statistical Society, Series B, 48, 244–248.MathSciNetMATHGoogle Scholar
  21. Hunsberger, S., Albert, P. S., Follmann, D. A., Suh, E. (2002). Parametric and semiparametric approaches to testing for seasonal trend in serial count data. Biostatistics, 3, 289–298.Google Scholar
  22. Knight, K. (1998). Limiting distributions for \(L_1\) regression estimators under general conditions. The Annals of Statistics, 26(2), 755–770.MathSciNetCrossRefMATHGoogle Scholar
  23. Leng, C., Liang, H., Martinson, N. (2011). Efficient variable selection for semiparametric generalized partially linear models with applications in study of condom use for HIV patients. Statistics in Medicine, 30, 2015–2027.Google Scholar
  24. Lian, H., Liang, H., Carroll, R. J. (2015). Variance function partially linear single-index models. Journal of the Royal Statistical Society, Series B, 77, 171–194.Google Scholar
  25. Liang, F., Paulo, R., Molina, G., Clyde, M. A., Berger, J. O. (2008). Mixtures of \(g\) priors for Bayesian variable selection. Journal of the American Statistical Association, 103(481), 410–423.Google Scholar
  26. Liang, H., Härdle, W., Carroll, R. (1999). Estimation in a semiparametric partially linear errors-in-variables model. The Annals of Statistics, 27, 1519–1535.Google Scholar
  27. Liang, H., Liu, X., Li, R., Tsai, C. L. (2010). Estimation and testing for partially linear single-index models. The Annals of Statistics, 38, 3811–3836.Google Scholar
  28. Ma, Y., Zhu, L. (2012). Doubly robust and efficient estimators for heteroscedastic partially linear single-index models allowing high dimensional covariates. Journal of the Royal Statistical Society: Series B, 75, 305–322.Google Scholar
  29. Ma, Y., Chiou, J.-M., Wang, N. (2006). Efficient semiparametric estimator for heteroscedastic partially linear models. Biometrika, 93, 75–84.Google Scholar
  30. Newey, W. K. (1994). The asymptotic variance of semiparametric estimators. Econometrica, 62, 1349–1382.MathSciNetCrossRefMATHGoogle Scholar
  31. Opsomer, J., Ruppert, D. (1997). Fitting a bivariate additive model by local polynomial regression. The Annals of Statistics, 25, 186–211.Google Scholar
  32. Prada-Sánchez, J., Febrero-Bande, M., Cotos-Yáñez, T., González-Manteiga, W., Bermúdez-Cela, J., Lucas-Dominguez, T. (2000). Prediction of \(\text{so}_2\) pollution incidents near a power station using partially linear models and an historical matrix of predictor-response vectors. Environmetrics, 11, 209–225.Google Scholar
  33. Robinson, P. M. (1988). Root \(n\)-consistent semiparametric regression. Econometrica, 56, 931–954.MathSciNetCrossRefMATHGoogle Scholar
  34. Severini, T. A., Staniswalis, J. G. (1994). Quasi-likelihood estimation in semiparametric models. Journal of the American Statistical Association, 89, 501–511.Google Scholar
  35. Speckman, P. E. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B, 50, 413–436.MathSciNetMATHGoogle Scholar
  36. Teschendorff, A. E., Widschwendter, M. (2012). Differential variability improves the identification of cancer risk markers in dna methylation studies profiling precursor cancer lesions. Bioinformatics, 28, 1487–1494.Google Scholar
  37. Thomas, L., Stefanski, L. A., Davidian, M. (2012) . Measurement error model methods for bias reduction and variance estimation in logistic regression with estimated variance predictors. Technical report, North Carolina State University.Google Scholar
  38. Wang, H., Leng, C. (2007). Unified lasso estimation via least squares approximation. Journal of the American Statistical Association, 101, 1418–1429.Google Scholar
  39. Wang, L., Liu, X., Liang, H., Carroll, R. (2011). Estimation and variable selection for generalized additive partial linear models. The Annals of Statistics, 39, 1827–1851.Google Scholar
  40. Western, B., Bloome, D. (2009). Variance function regressions for studying inequality. Sociological Methodology, 39, 293–326.Google Scholar
  41. Xia, Y. C., Härdle, W. (2006). Semi-parametric estimation of partially linear single-index models. Journal of Multivariate Analysis, 97, 1162–1184.Google Scholar
  42. Yatchew, A., No, J. A. (2001). Household gasoline demand in Canada. Econometrica, 69(6), 1697–1709.Google Scholar
  43. Zeger, S., Diggle, P. (1994). Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters. Biometrics, 50, 689–699.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Jersey Institute of TechnologyUniversity Heights, NewarkUSA
  2. 2.Department of MathematicsCity University of Hong KongKowloon TongHong Kong
  3. 3.Department of StatisticsThe George Washington UniversityWashingtonUSA

Personalised recommendations