未知误差分布下线性回归模型的非参数自适应估计
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摘要
对于正态分布误差,线性回归模型的极大似然估计(Maximum likelihood estimate,MLE)与最小二乘估计(Least squares estimate,LSE)是等价的.当高斯性假设不成立时,MLE比LSE更有效.然而,当误差分布未知时,MLE通常是不可实现的.文中给出了未知误差分布下线性回归模型系数的非参数自适应估计,证明了估计量渐近有效于已知误差分布下线性回归模型系数的MLE,并给出了回归系数的一个轮廓似然比检验统计量.
For normally distributed errors,the maximum likelihood estimate(MLE)is equivalent to the least squares estimate(LSE)in linear regression models.In the absence of Gaussianity,MLE is more effective than LSE.However,the error distribution is generally unknown,and MLE is infeasible.Anonparametric adaptive method is proposed to estimate parameters in a linear regression model with unknown error distribution,the resulting estimator is asymptotically as efficient as the oracle MLE that the error distribution is known.A profile likelihood ratio test for regression parameters is also proposed.
For normally distributed errors,the maximum likelihood estimate(MLE)is equivalent to the least squares estimate(LSE)in linear regression models.In the absence of Gaussianity,MLE is more effective than LSE.However,the error distribution is generally unknown,and MLE is infeasible.Anonparametric adaptive method is proposed to estimate parameters in a linear regression model with unknown error distribution,the resulting estimator is asymptotically as efficient as the oracle MLE that the error distribution is known.A profile likelihood ratio test for regression parameters is also proposed.
引文
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[2]LIU Min-hui,BOZDOGAN H.Multivariate regression models with power exponential random errors and subset selection using genetic algorithms with information complexity[J].European Journal of Pure and Applied Mathematics,2008,1(1):4.
[3]ISLAM M Q,TIKU M L.Multiple linear regression model with stochastic design variables[J].Journal of Applied Statistics,2010,37(6):923.
[4]JAHAN S,KHAN A.Power of t-test for simple linear regression model with non-normal error distribution:aquantile function distribution approach[J].Journal of Scientific Research,2012,4(3):609.
[5]ATSEDEWEYN A A,SRINIVASA R K.Linear regression model with new symmetric distributed errors[J].Journal of Applied Statistics,2014,41(2):364.
[6]STONE C J.Adaptive maximum likelihood estimators of a location parameter[J].Annals of Statistics,1975,3(2):267.
[7]KOUL H L,SUSARLA V.Adaptive estimation in linear regression[J].Statist&Decisions,1983,1(4/5):379.
[8]STEIGERWALD D G.Adaptive estimation in time series regression models[J].Journal of Econometrics,1992,54(1):251.
[9]LINTON O.Adaptive estimation in ARCH models[J].Econometric Theory,1993,9(4):539.
[10]KOUL H L,SCHICK A.Efficient estimation in nonlinear autore-gressive time-series models[J].Bernoulli,1997,3(3):247.
[11]SCHICK A.On efficient estimation in regression models[J].The Annals of Statistics,1993,21(3):1486.
[12]SCHICK A.An adaptive estimator of the autocorrelation coefficient in regression models with autoregressive errors[J].Journal of Time Series Analysis,1998,19(5):575.
[13]HODGSON D J,LINTON O,VORKINK K.Testing the capital asset pricing model efficiently under elliptical symmetry:a semiparametric approach[J].Journal of Applied Econometrics,2002,17(6):617.
[14]LINTON O,XIAO Zhi-jie.A nonparametric regression estimator that adapts to error distribution of unknown form[J].Econometric Theory,2007,23(3):371.
[15]YUAN Ao.Semiparametric inference with kernel likelihood[J].Journal of Nonparametric Statistics,2009,21(2):207.
[16]YUAN Ao,DE GOOIJER J G.Semiparametric regression with kernel error model[J].Scandinavian Journal of Statistics,2007,34(4):841.
[17]WANG Qin,YAO Wei-xin.An adaptive estimation of MAVE[J].Journal of Multivariate Analysis,2012,104(1):88.
[18]YAO Wei-xin,ZHAO Zhi-biao.Kernel densitybased linear regression estimate[J].Communications in Statistics:Theory and Methods,2013,42(23):4499.
[19]HUANG A-lan.Density estimation and nonparametric inferences using maximum likelihood weighted kernels[J].Journal of Nonparametric Statistics,2013,25(3):561.
[20]JONES M C,HENDERSON D A.Maximum likelihood kernel density estimation:on the potential of convolution sieves[J].Computational Statistics and Data Analysis,2009,53(10):3726.
[21]AI Chun-rong.A semiparametric maximum likelihood estimator[J].Econometrica,1997,65(4):933.
[22]SILVERMAN B W.Choosing the window width when estimating a density[J].Biometrika,1978,65(1):1.
[23]MURPHY S A,VAN DER VAART A W.Semiparametric mixtures in case-control studies[J].Journal of Multivariate Analysis,2001,79(1):1.
[24]HALL P,TURLACH B A.Reducing bias in curve estimation by use of weights[J].Computational Statistics&Data Analysis,1999,30(1):67.
[25]MURPHY S A,VAN DER VAART A W.On profile likelihood(with comments and rejoinder by the authors)[J].Journal of the American Statistical Association,2000,95(450):449.
[26]MARRON J S,WAND M P.Exact mean integrated squared error[J].Annals of Statistics,1992,20(2):712.