The Annals of Statistics

Maximum likelihood estimation in transformed linear regression with nonnormal errors

Xingwei Tong, Fuqing Gao, Kani Chen, Dingjiao Cai, and Jianguo Sun

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Abstract

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.

Article information

Source
Ann. Statist., Volume 47, Number 4 (2019), 1864-1892.

Dates
Received: August 2016
Revised: May 2018
First available in Project Euclid: 21 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1558425633

Digital Object Identifier
doi:10.1214/18-AOS1726

Mathematical Reviews number (MathSciNet)
MR3953438

Zentralblatt MATH identifier
07082273

Subjects
Primary: 62F12: Asymptotic properties of estimators

Keywords
Linear transformation model maximum likelihood estimation

Citation

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


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Supplemental materials

  • 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.