Iterative weighted estimation based on variance modelling in linear regression models

ABSTRACT

The estimation of variance function plays an extremely important role in statistical inference of the regression models. In this paper we propose a variance modelling method for constructing the variance structure via combining the exponential polynomial modelling method and the kernel smoothing technique. A simple estimation method for the parameters in heteroscedastic linear regression models is developed when the covariance matrix is unknown diagonal and the variance function is a positive function of the mean. The consistency and asymptotic normality of the resulting estimators are established under some mild assumptions. In particular, a simple version of bootstrap test is adapted to test misspecification of the variance function. Some Monte Carlo simulation studies are carried out to examine the finite sample performance of the proposed methods. Finally, the methodologies are illustrated by the ozone concentration dataset.

KEYWORDS:

• Asymptotic properties
• Linear regression models
• Modified Levenberg-Marquardt method
• Variance modelling

MATHEMATICS SUBJECT CLASSIFICATION:

• primary 62G05
• secondary 62G07

Acknowledgements

The authors are grateful to the Editor and two anonymous referees for their constructive comments which have greatly improved this paper.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China under Grants No. 11571073, 11701286, the National Social Science Foundation under Grant No. 17CTJ016, the Natural Science Foundation of Jiangsu Province of China under Grant No. BK20171073, and the Philosophy and Social Science Foundation of the Jiangsu Higher Education Institutions of China under Grant No. 2017SJB0350.