基于最小化复合分位损失函数的尺度参数估计和异质性检验
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摘要
在进行回归分析时,对误差项离散程度的度量是一个重要话题.文章利用最小化复合分位损失的方法,对误差项的尺度参数进行估计,并证明估计量的大样本性质.进一步的研究表明:通过选取合适的分位数,能得到尺度参数的最优估计,并以此进行异质性检验.模拟结果表明,在重尾条件下所提出的方法有更高的精度.实际数据应用体现了该方法的良好性能.
When it comes to regression analysis, the measurement of residuals' scale is an important issue. In this paper, we propose an estimator of the scale-parameter of residuals by minimizing the composite quantile loss function and prove its large sample properties. Further research shows that by selecting appropriate quantiles,we could optimize the proposed estimator. The heteroscedasticity is tested based on the resulting estimators results. Simulation shows that the method is more effective in estimating the scale-parameter and testing heteroscedasticity compared with other methods under the setting of heavy-tailed distributions. Applications of this method to real data show its good performance.
When it comes to regression analysis, the measurement of residuals' scale is an important issue. In this paper, we propose an estimator of the scale-parameter of residuals by minimizing the composite quantile loss function and prove its large sample properties. Further research shows that by selecting appropriate quantiles,we could optimize the proposed estimator. The heteroscedasticity is tested based on the resulting estimators results. Simulation shows that the method is more effective in estimating the scale-parameter and testing heteroscedasticity compared with other methods under the setting of heavy-tailed distributions. Applications of this method to real data show its good performance.
引文
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[3]Breusch T S, Pagan A R. A simple test for heteroscedasticity and random coefficient variation.Econometrica, 1979, 47(5):1287-1294.
[4]Cook R D, Weisberg S. Diagnostics for heteroscedasticity in regression. Biometrika, 1983, 70(1):1-10.
[5]Davidian M, Carroll R J. Variance function estimation. Journal of the American tatistical Association, 1987, 82(400):1079-1091.
[6]M(u|¨)ller H G,Stadtm(u|¨)ller U. Estimation of heteroscedasticity in regression analysis. The Annals of Statistics, 1987, 15(2):610-625.
[7]Eubank R L, Thomas W. Detecting heteroscedasticity in nonparametric regression. Journal of the Royal Statistical Society, Series B(Methodological), 1993, 55(1):145-155.
[8]Dette H, Munk A. Testing heteroscedasticity in nonparametric regression. Journal of the Royal Statistical Society, Series B(Statistical Methodology), 1998, 60(4):693-708.
[9]Koenker R, Bassett G. Robust tests for heteroscedasticity based on regression quantiles. Econometrica, 1982, 50(1):43-61.
[10]Zou H, Yuan M. Composite quantile regression and the oracle model selection theory. The Annals of Statistics, 2008, 36(3):1108-1126.
[11]Xia W T, Xiong W, Tian M Z. Heteroscedasticity detection and estimation with quantile difference method. Journal of Systems Science and Complexity, 2016, 29(2):511-530.
[12]Hart J D. Nonparametric smoothing and lack-of-fit tests. Technometrics, 1997, 41(2):175-176.
[13]Von Neumann J. Distribution of the ratio of the mean square successive difference to the variance.The Annals of Mathematical Statistics, 1941, 12(4):367-395.
[14]Von Neumann J, Kent R H, Bellinson H R, et al. The mean square successive difference. The Annals of Mathematical Statistics, 1941, 12(2):153-162.
[15]Bailar B. Salary survey of U.S. colleges and universities offering degrees in statistics. Amstat News, 1991, 182:3-10.
[16]Knight K. Limiting distributions for L1 regression estimators under general conditions. Annals of Statistics, 1998, 26(2):755-770.
[2]Harrison M J, McCabe P M. A test for heteroscedasticity based on ordinary least squares residuals. Journal of the American Statistical Association, 1979, 74(366):494-499.
[3]Breusch T S, Pagan A R. A simple test for heteroscedasticity and random coefficient variation.Econometrica, 1979, 47(5):1287-1294.
[4]Cook R D, Weisberg S. Diagnostics for heteroscedasticity in regression. Biometrika, 1983, 70(1):1-10.
[5]Davidian M, Carroll R J. Variance function estimation. Journal of the American tatistical Association, 1987, 82(400):1079-1091.
[6]M(u|¨)ller H G,Stadtm(u|¨)ller U. Estimation of heteroscedasticity in regression analysis. The Annals of Statistics, 1987, 15(2):610-625.
[7]Eubank R L, Thomas W. Detecting heteroscedasticity in nonparametric regression. Journal of the Royal Statistical Society, Series B(Methodological), 1993, 55(1):145-155.
[8]Dette H, Munk A. Testing heteroscedasticity in nonparametric regression. Journal of the Royal Statistical Society, Series B(Statistical Methodology), 1998, 60(4):693-708.
[9]Koenker R, Bassett G. Robust tests for heteroscedasticity based on regression quantiles. Econometrica, 1982, 50(1):43-61.
[10]Zou H, Yuan M. Composite quantile regression and the oracle model selection theory. The Annals of Statistics, 2008, 36(3):1108-1126.
[11]Xia W T, Xiong W, Tian M Z. Heteroscedasticity detection and estimation with quantile difference method. Journal of Systems Science and Complexity, 2016, 29(2):511-530.
[12]Hart J D. Nonparametric smoothing and lack-of-fit tests. Technometrics, 1997, 41(2):175-176.
[13]Von Neumann J. Distribution of the ratio of the mean square successive difference to the variance.The Annals of Mathematical Statistics, 1941, 12(4):367-395.
[14]Von Neumann J, Kent R H, Bellinson H R, et al. The mean square successive difference. The Annals of Mathematical Statistics, 1941, 12(2):153-162.
[15]Bailar B. Salary survey of U.S. colleges and universities offering degrees in statistics. Amstat News, 1991, 182:3-10.
[16]Knight K. Limiting distributions for L1 regression estimators under general conditions. Annals of Statistics, 1998, 26(2):755-770.