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Robust Estimation Procedure for Autoregressive Models with Heterogeneity

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Abstract

In environmental studies, regression analysis is frequently performed. The classical approach is the ordinary least squares method which consists in minimizing the sum of the squares of the residuals. However, this method relies on strong assumptions that are not always satisfied. In environmental data, the response variable often contains outliers and errors can be heteroscedastic. This can have significant effects on parameter estimation. To solve this problem, the weighted M-estimation was developed. It assumes a parametric function for the variance, and, estimates alternately and robustly, mean and variance parameters. However, this method is limited to the independent errors case, and is not applicable to time series data. Therefore, we introduce a new estimation procedure which adapts the weighted M-estimation to environmental time series data, while selecting optimal value for the tuning parameter present in the M-estimation. We compare the efficiency of our procedure on simulated data to other usual regression methods. Our estimation procedure outperforms the other methods providing estimates with lower biases and mean squared errors. Finally, we illustrate the proposed method using an air quality dataset from Beijing. This method has been implemented in the R package RlmDataDriven.

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Acknowledgements

The work was carried out during the visit of Aurelien Callens to the School of Mathematical Science, Queensland University of Technology, Brisbane, Australia.

Funding

This research was partially funded by the Australian Research Council project (DP160104292). Liya Fu’s research was supported by the National Natural Science Foundation of China (11871390), the Fundamental Research Funds for the Central Universities (No. xjj2017180), the Natural Science Basic Research Plan in Shaanxi Province of China (2018JQ1006).

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Authors and Affiliations

  1. Laboratoire de Mathématiques et de leurs Applications de Pau, Université de Pau et des Pays de l’Adour, UMR CNRS 5142, E2S-UPPA, Pau, France

    A. Callens & B. Liquet

  2. ARC Centre of Excellence for Mathematical and Statistical Frontiers and School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia

    Y.-G. Wang

  3. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China

    L. Fu

Authors
  1. A. Callens
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  2. Y.-G. Wang
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  3. L. Fu
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  4. B. Liquet
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Correspondence to A. Callens.

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Callens, A., Wang, YG., Fu, L. et al. Robust Estimation Procedure for Autoregressive Models with Heterogeneity. Environ Model Assess 26, 313–323 (2021). https://doi.org/10.1007/s10666-020-09730-w

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