The Annals of Statistics
- Ann. Statist.
- Volume 47, Number 4 (2019), 1864-1892.
Maximum likelihood estimation in transformed linear regression with nonnormal errors
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.
Ann. Statist., Volume 47, Number 4 (2019), 1864-1892.
Received: August 2016
Revised: May 2018
First available in Project Euclid: 21 May 2019
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62F12: Asymptotic properties of estimators
Tong, Xingwei; Gao, Fuqing; Chen, Kani; Cai, Dingjiao; Sun, Jianguo. Maximum likelihood estimation in transformed linear regression with nonnormal errors. Ann. Statist. 47 (2019), no. 4, 1864--1892. doi:10.1214/18-AOS1726. https://projecteuclid.org/euclid.aos/1558425633
- Supplement to “Maximum likelihood estimation in transformed linear regression with nonnormal errors”. Due to space constraints, the proofs of the consistency of the proposed covariance matrix estimator and the identifiability of model (1.1) along with some additional simulation results are provided in the Supplementary Material.