Maximum likelihood estimation in transformed linear regression with nonnormal errors. (English) Zbl 1421.62022

Summary: This paper discusses the transformed linear regression with non-normal error distributions, a problem that often occurs in many areas such as economics and social sciences as well as medical studies. The linear transformation model is an important tool in survival analysis partly due to its flexibility. In particular, it includes the Cox model and the proportional odds model as special cases when the error follows the extreme value distribution and the logistic distribution, respectively. Despite the popularity and generality of linear transformation models, however, there is no general theory on the maximum likelihood estimation of the regression parameter and the transformation function. One main difficulty for this is that the transformation function near the tails diverges to infinity and can be quite unstable. It affects the accuracy of the estimation of the transformation function and regression parameters. In this paper, we develop the maximum likelihood estimation approach and provide the near optimal conditions on the error distribution under which the consistency and asymptotic normality of the resulting estimators can be established. Extensive numerical studies suggest that the methodology works well, and an application to the data on a typhoon forecast is provided.


62F12 Asymptotic properties of parametric estimators
62J05 Linear regression; mixed models
62P12 Applications of statistics to environmental and related topics
Full Text: DOI Euclid


[1] Box, G. E. P. and Cox, D. R. (1964). An analysis of transformations (with discussion). J. Roy. Statist. Soc. Ser. B26 211-252. · Zbl 0156.40104 · doi:10.1111/j.2517-6161.1964.tb00553.x
[2] Chen, K., Jin, Z. and Ying, Z. (2002). Semiparametric analysis of transformation models with censored data. Biometrika89 659-668. · Zbl 1039.62094 · doi:10.1093/biomet/89.3.659
[3] Chen, K. and Tong, X. (2010). Varying coefficient transformation models with censored data. Biometrika97 969-976. · Zbl 1204.62128 · doi:10.1093/biomet/asq032
[4] Chen, S. (2002). Rank estimation of transformation models. Econometrica70 1683-1697. · Zbl 1101.62020 · doi:10.1111/1468-0262.00347
[5] Cheng, S. C., Wei, L. J. and Ying, Z. (1995). Analysis of transformation models with censored data. Biometrika82 835-845. · Zbl 0861.62071 · doi:10.1093/biomet/82.4.835
[6] Cox, D. R. (1972). Regression models and life-tables. J. Roy. Statist. Soc. Ser. B34 187-220. · Zbl 0243.62041 · doi:10.1111/j.2517-6161.1972.tb00899.x
[7] Diao, G., Zeng, D. and Yang, S. (2013). Efficient semiparametric estimation of short-term and long-term hazard ratios with right-censored data. Biometrics69 840-849. · Zbl 1291.62176 · doi:10.1111/biom.12097
[8] Doksum, K. A. (1987). An extension of partial likelihood methods for proportional hazard models to general transformation models. Ann. Statist.15 325-345. · Zbl 0639.62026 · doi:10.1214/aos/1176350269
[9] Fine, J. P., Ying, Z. and Wei, L. J. (1998). On the linear transformation model with censored observations. Biometrika85 980-986. · Zbl 0921.62033 · doi:10.1093/biomet/85.4.980
[10] Han, A. K. (1987). A nonparametric analysis of transformations. J. Econometrics35 191-209. · Zbl 0649.62037 · doi:10.1016/0304-4076(87)90023-6
[11] Horowitz, J. L. (1996). Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. Econometrica64 103-137. · Zbl 0861.62029 · doi:10.2307/2171926
[12] Huang, J. (1999). Efficient estimation of the partly linear additive Cox model. Ann. Statist.27 1536-1563. · Zbl 0977.62035 · doi:10.1214/aos/1017939141
[13] Khan, S. and Tamer, E. (2007). Partial rank estimation of duration models with general forms of censoring. J. Econometrics136 251-280. · Zbl 1418.62369 · doi:10.1016/j.jeconom.2006.03.003
[14] Ma, S. and Kosorok, M. R. (2005). Penalized log-likelihood estimation for partly linear transformation models with current status data. Ann. Statist.33 2256-2290. · Zbl 1086.62056 · doi:10.1214/009053605000000444
[15] Murphy, S. A. (1994). Consistency in a proportional hazards model incorporating a random effect. Ann. Statist.22 712-731. · Zbl 0827.62033 · doi:10.1214/aos/1176325492
[16] Murphy, S. A. (1995). Asymptotic theory for the frailty model. Ann. Statist.23 182-198. · Zbl 0822.62069 · doi:10.1214/aos/1176324462
[17] Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood. J. Amer. Statist. Assoc.95 449-485. · Zbl 0995.62033 · doi:10.1080/01621459.2000.10474219
[18] Sherman, R. P. (1993). The limiting distribution of the maximum rank correlation estimator. Econometrica61 123-137. · Zbl 0773.62011 · doi:10.2307/2951780
[19] Tong, X., Gao, F., Chen, K., Cai, D. and Sun, J. (2019). Supplement to “Maximum likelihood estimation in transformed linear regression with nonnormal errors.” DOI:10.1214/18-AOS1726SUPP. · Zbl 1421.62022
[20] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics3. Cambridge Univ. Press, Cambridge. · Zbl 0910.62001
[21] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York. · Zbl 0862.60002
[22] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Application to Statistics. Springer Series in Statistics. Springer, New York. · Zbl 0862.60002
[23] Zeng, D. and Lin, D. Y. (2006). Maximum likelihood estimation in semiparametric transformation models for counting processes. Biometrika93 627-640. · Zbl 1108.62083 · doi:10.1093/biomet/93.3.627
[24] Zeng, D. and Lin, D. Y. (2007). Maximum likelihood estimation in semiparametric regression models with censored data (with discussion). J. R. Stat. Soc. Ser. B. Stat. Methodol.69 507-564. · Zbl 07555364
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.