Robust Parameter Estimation in PLMs with Skewed and Heavy-tailed Error Distributions

Fatma Zehra Doğru; Olcay Arslan

doi:10.31801/cfsuasmas.1677165

Robust Parameter Estimation in PLMs with Skewed and Heavy-tailed Error Distributions

Abstract

Partially linear models (PLMs) provide a flexible extension of classical linear models by incorporating parametric and nonparametric components, making them suitable for complex data structures often encountered in economics and biometrics. Traditionally, these models assume normally distributed errors, an assumption that may not hold in many practical scenarios. To address this, previous studies introduced the skew normal distribution in PLMs (PLM-SN) to account for asymmetry. However, such data sets may also exhibit heavy tails and skewness. To capture both characteristics, we propose a new approach: modeling the error structure using the skewed Laplace normal distribution (PLM-SLN), which maintains the same number of parameters as PLM-SN but better accommodates heavy-tailed behavior. An efficient estimation procedure using an expectation-conditional maximization (ECM) algorithm is developed to compute maximum likelihood estimates. The proposed method is validated using simulation experiments and applied to an empirical dataset to highlight its practical utility.

Keywords

References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petrov, B. N. and Caski, F., editors, Proceeding of the Second International Symposium on Information Theory, pages 267–281. Akademiai Kiado.
Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, pages 171–178.
Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistics, 46(2):199–208.
Bazrafkan, M., Zare, K., Maleki, M., and Khodadi, Z. (2022). Partially linear models based on heavy-tailed and asymmetrical distributions. Stochastic Environmental Research and Risk Assessment, 36(5):1243–1253.
Burnham, K. and Anderson, D. (2002). Model Selection and Multi-Model Inference. Springer, New York.
DiCiccio, T. J. and Monti, A. C. (2004). Inferential aspects of the skew exponential power distribution. Journal of the American Statistical Association, 99(466):439–450.
Doğru, F. Z. and Arslan, O. (2017). Parameter estimation for mixtures of skew Laplace normal distributions and application in mixture regression modeling. Communications in Statistics – Theory and Methods, 46(21):10879–10896.
Doğru, F. Z. and Arslan, O. (2019). Joint modelling of the location, scale and skewness parameters of the skew Laplace normal distribution. Iranian Journal of Science and Technology, Transactions A: Science, 43(3):1249–1257.

Doğru, F. Z. and Arslan, O. (2021). Finite mixtures of skew Laplace normal distributions with random skewness. Computational Statistics, 36(1):423–447.
Doğru, F. Z., Yu, K., and Arslan, O. (2019). Heteroscedastic and heavy-tailed regression with mixtures of skew Laplace normal distributions. Journal of Statistical Computation and Simulation, 89(17):3213– 3240.
Ferreira, C. S., Borelli Zeller, C., and de Oliveira Garcia, R. R. (2023). Heteroscedastic partially linear model under skew-normal distribution with application in ragweed pollen concentration. Journal of Applied Statistics, 50(6):1255–1282.
Ferreira, C. S. and Paula, G. A. (2017). Estimation and diagnostic for skew-normal partially linear models. Journal of Applied Statistics, 44(16):3033–3053.
Ferreira, C. S., Paula, G. A., and Lana, G. C. (2022). Estimation and diagnostic for partially linear models with first-order autoregressive skew-normal errors. Computational Statistics, 37(1):445–468.
Gómez, H. W., Venegas, O., and Bolfarine, H. (2007). Skew-symmetric distributions generated by the distribution function of the normal distribution. Environmetrics, 18(4):395–407.
Green, P. J. (1987). Penalized likelihood for general semi-parametric regression models. International Statistical Review, pages 245–259.
Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman and Hall, Boca Raton.
Hastie, T. and Tibshirani, R. (1990). Generalized Additive Models. Monographs on Statistics & Applied Probability. Chapman and Hall/CRC.
Ibacache-Pulgar, G. and Paula, G. A. (2011). Local influence for student t partially linear models. Computational Statistics & Data Analysis, 55(3):1462–1478.
Ibacache-Pulgar, G., Paula, G. A., and Cysneiros, F. J. A. (2013). Semiparametric additive models under symmetric distributions. TEST, 22(1):103–121.
Maleki, M. and Mahmoudi, M. R. (2017). Two-piece location-scale distributions based on scale mixtures of normal family. Communications in Statistics – Theory and Methods, 46(24):12356–12369.
Meng, X.-L. and Rubin, D. B. (1993). Maximum likelihood estimation via the ecm algorithm: A general framework. Biometrika, 80(2):267–278.
Relvas, C. E. M. and Paula, G. A. (2016). Partially linear models with first-order autoregressive symmetric errors. Statistical Papers, 57(3):795–825.
Ruppert, D., Wand, M. P., and Carroll, R. (2003). Semiparametric Regression. Cambridge University Press, New York.
Stark, P. C., Ryan, L. M., McDonald, J. L., and Burge, H. A. (1997). Using meteorologic data to predict daily ragweed pollen levels. Aerobiologia, 13(3):177–184.
Vanegas, L. H. and Paula, G. A. (2015). A semiparametric approach for joint modeling of median and skewness. TEST, 24:110–135.
Zhu, H.-T. and Lee, S.-Y. (2001). Local influence for incomplete data models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63(1):111–126.

Details

Primary Language

English

Subjects

Computational Statistics, Statistical Theory

Journal Section

Research Article

Authors

Fatma Zehra Doğru ^*
0000-0001-8220-2375
Türkiye

Olcay Arslan
0000-0002-7067-4997
Türkiye

Publication Date

June 30, 2026

Submission Date

April 22, 2025

Acceptance Date

September 9, 2025

Published in Issue

Year 2026 Volume: 75 Number: 2

DOI

https://doi.org/10.31801/cfsuasmas.1677165

IZ

https://izlik.org/JA75AY99MR

Cite

RIS / Bibtex

APA

Doğru, F. Z., & Arslan, O. (2026). Robust Parameter Estimation in PLMs with Skewed and Heavy-tailed Error Distributions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 75(2), 211-227. https://doi.org/10.31801/cfsuasmas.1677165

AMA

1.Doğru FZ, Arslan O. Robust Parameter Estimation in PLMs with Skewed and Heavy-tailed Error Distributions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2026;75(2):211-227. doi:10.31801/cfsuasmas.1677165

Chicago

Doğru, Fatma Zehra, and Olcay Arslan. 2026. “Robust Parameter Estimation in PLMs With Skewed and Heavy-Tailed Error Distributions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 75 (2): 211-27. https://doi.org/10.31801/cfsuasmas.1677165.

EndNote

Doğru FZ, Arslan O (June 1, 2026) Robust Parameter Estimation in PLMs with Skewed and Heavy-tailed Error Distributions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 75 2 211–227.

IEEE

[1]F. Z. Doğru and O. Arslan, “Robust Parameter Estimation in PLMs with Skewed and Heavy-tailed Error Distributions”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 75, no. 2, pp. 211–227, June 2026, doi: 10.31801/cfsuasmas.1677165.

ISNAD

Doğru, Fatma Zehra - Arslan, Olcay. “Robust Parameter Estimation in PLMs With Skewed and Heavy-Tailed Error Distributions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 75/2 (June 1, 2026): 211-227. https://doi.org/10.31801/cfsuasmas.1677165.

JAMA

1.Doğru FZ, Arslan O. Robust Parameter Estimation in PLMs with Skewed and Heavy-tailed Error Distributions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2026;75:211–227.

MLA

Doğru, Fatma Zehra, and Olcay Arslan. “Robust Parameter Estimation in PLMs With Skewed and Heavy-Tailed Error Distributions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 75, no. 2, June 2026, pp. 211-27, doi:10.31801/cfsuasmas.1677165.

Vancouver

1.Fatma Zehra Doğru, Olcay Arslan. Robust Parameter Estimation in PLMs with Skewed and Heavy-tailed Error Distributions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2026 Jun. 1;75(2):211-27. doi:10.31801/cfsuasmas.1677165