Robust-BD Estimation and Inference for General Partially Linear Models
The classical quadratic loss for the partially linear model (PLM) and the likelihood function for the generalized PLM are not resistant to outliers. This inspires us to propose a class of “robust-Bregman divergence (BD)” estimators of both the parametric and nonparametric components in the general p...
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| Format: | Article |
| Language: | English |
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MDPI AG
2017-11-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/19/11/625 |
| _version_ | 1811299906558623744 |
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| author | Chunming Zhang Zhengjun Zhang |
| author_facet | Chunming Zhang Zhengjun Zhang |
| author_sort | Chunming Zhang |
| collection | DOAJ |
| description | The classical quadratic loss for the partially linear model (PLM) and the likelihood function for the generalized PLM are not resistant to outliers. This inspires us to propose a class of “robust-Bregman divergence (BD)” estimators of both the parametric and nonparametric components in the general partially linear model (GPLM), which allows the distribution of the response variable to be partially specified, without being fully known. Using the local-polynomial function estimation method, we propose a computationally-efficient procedure for obtaining “robust-BD” estimators and establish the consistency and asymptotic normality of the “robust-BD” estimator of the parametric component
β
o
. For inference procedures of
β
o
in the GPLM, we show that the Wald-type test statistic
W
n
constructed from the “robust-BD” estimators is asymptotically distribution free under the null, whereas the likelihood ratio-type test statistic
Λ
n
is not. This provides an insight into the distinction from the asymptotic equivalence (Fan and Huang 2005) between
W
n
and
Λ
n
in the PLM constructed from profile least-squares estimators using the non-robust quadratic loss. Numerical examples illustrate the computational effectiveness of the proposed “robust-BD” estimators and robust Wald-type test in the appearance of outlying observations. |
| first_indexed | 2024-04-13T06:43:34Z |
| format | Article |
| id | doaj.art-1ed624e4e07a4187adfa838b868056d4 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-13T06:43:34Z |
| publishDate | 2017-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-1ed624e4e07a4187adfa838b868056d42022-12-22T02:57:40ZengMDPI AGEntropy1099-43002017-11-01191162510.3390/e19110625e19110625Robust-BD Estimation and Inference for General Partially Linear ModelsChunming Zhang0Zhengjun Zhang1Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706, USADepartment of Statistics, University of Wisconsin-Madison, Madison, WI 53706, USAThe classical quadratic loss for the partially linear model (PLM) and the likelihood function for the generalized PLM are not resistant to outliers. This inspires us to propose a class of “robust-Bregman divergence (BD)” estimators of both the parametric and nonparametric components in the general partially linear model (GPLM), which allows the distribution of the response variable to be partially specified, without being fully known. Using the local-polynomial function estimation method, we propose a computationally-efficient procedure for obtaining “robust-BD” estimators and establish the consistency and asymptotic normality of the “robust-BD” estimator of the parametric component β o . For inference procedures of β o in the GPLM, we show that the Wald-type test statistic W n constructed from the “robust-BD” estimators is asymptotically distribution free under the null, whereas the likelihood ratio-type test statistic Λ n is not. This provides an insight into the distinction from the asymptotic equivalence (Fan and Huang 2005) between W n and Λ n in the PLM constructed from profile least-squares estimators using the non-robust quadratic loss. Numerical examples illustrate the computational effectiveness of the proposed “robust-BD” estimators and robust Wald-type test in the appearance of outlying observations.https://www.mdpi.com/1099-4300/19/11/625Bregman divergencegeneralized linear modellocal-polynomial regressionmodel checknonparametric testquasi-likelihoodsemiparametric modelWald statistic |
| spellingShingle | Chunming Zhang Zhengjun Zhang Robust-BD Estimation and Inference for General Partially Linear Models Entropy Bregman divergence generalized linear model local-polynomial regression model check nonparametric test quasi-likelihood semiparametric model Wald statistic |
| title | Robust-BD Estimation and Inference for General Partially Linear Models |
| title_full | Robust-BD Estimation and Inference for General Partially Linear Models |
| title_fullStr | Robust-BD Estimation and Inference for General Partially Linear Models |
| title_full_unstemmed | Robust-BD Estimation and Inference for General Partially Linear Models |
| title_short | Robust-BD Estimation and Inference for General Partially Linear Models |
| title_sort | robust bd estimation and inference for general partially linear models |
| topic | Bregman divergence generalized linear model local-polynomial regression model check nonparametric test quasi-likelihood semiparametric model Wald statistic |
| url | https://www.mdpi.com/1099-4300/19/11/625 |
| work_keys_str_mv | AT chunmingzhang robustbdestimationandinferenceforgeneralpartiallylinearmodels AT zhengjunzhang robustbdestimationandinferenceforgeneralpartiallylinearmodels |