非参数半参数模型

非参数与半参数模型Thebasicideaofnonparametricinferenceistousedatatoinferanunknownquantitywhilemakingasfewassumptionsaspossible.0.00.20.40.60.81.00.00.51.0ty0.00.20.40.60.81.00.00.51.0ty主要内容核密度估计局部方法核回归局部线性回归变系数回归半变系数回归部分变系数回归全局方法样条回归多元非参数模型Cornwell与Rupert数据如何刻画随机变量的特征？女性（对数）工资男性（对数工资）Min.1stQu.MedianMean3rdQu.Max.4.6055.9586.2616.2556.5627.279Min.1stQu.MedianMean3rdQu.Max.5.0176.4586.7456.7306.9768.537直方图随机样本：x1,x2,…,xn直方图的构造确定原点x0，将数轴分割为宽度为h的区间(bin)数出落在每个区间的观察值个数，记为nj用nj除以n，再除以h，得到对每个区间，绘制高为fj，宽为h的柱形图jjhxhjxBj],,)1([00nhnfjj直方图中密度的一般表示nijjjihBxIBxInhxf1)()(1)(如何理解这个密度估计？有什么问题？如何改进？LWAGE的直方图HistogramoffwfwDensity4.55.05.56.06.57.07.50.00.6HistogramofmwmwDensity5.05.56.06.57.07.58.08.50.00.40.8HistogramoffwfwDensity4.55.05.56.06.57.00.01.02.0HistogramofmwmwDensity5.05.56.06.57.07.58.08.50.01.0HistogramoffwfwDensity4.55.05.56.06.57.0050150HistogramofmwmwDensity5.05.56.06.57.07.58.08.504080h对直方图的影响是什么？直方图的统计性质无偏？)()(1)(ˆ)(1)(ˆ)(1)()()(1)(1)(1)(ˆ)(1)(ˆ],,)1[(.0)1()1()1(110xfduufhxfBiasduufhxfEduufBxIPBxIEBxIEhBxInEnhBxIEnhxfEBxInhxfjhhjBxxjhhjhjhhjhjhhjjijijijinijihnijihj直方图的密度估计为：对于某个点假定原点为直方图密度估计的近似偏误))(()()(ˆ))(()()()21()()()()(1)()(1)1()1(xmmfxfxfExumfxfufhjmxfufduxfufhxfduufhjjhjjjhhjjhhj从而有：为处的一阶泰勒近似展开在直方图密度估计的偏误受什么影响？直方图密度估计的方差)(1)(1)(1)(1)(1)(ˆ22221xfnhduufduufnhnBxInVarhnBxInhVarxfVarjjBBjinijih直方图密度估计的方差受什么影响？直方图密度估计的均方误差逐点均方误差积分均方误差)1()(2121)(1)(ˆ)(ˆ)(ˆ222nhohoxhjhjfxfnhxfbiasxfVarxfMSEhhh22222222121)(1212121)()(1)(ˆ)()(ˆ)(ˆfhnhdxxfhnhdxhjfxhjBxIdxxfnhdxxfMSEdxxfxfExfMISEjjhhh最优带宽3/13/1220222222~60611)ˆ(121)ˆ(nfnhfhnhhfAMISEfhnhfAMISEhh对于标准正态分布，h0≈3.5n-1/3原点的影响fwDensity4.55.05.56.06.57.07.50.00.20.40.60.81.01.2x0=4.51fwDensity4.55.05.56.06.57.07.50.00.20.40.60.8x0=4.52fwDensity4.55.05.56.06.57.07.50.00.20.40.60.8x0=4.53fwDensity4.55.05.56.06.57.07.50.00.20.40.60.81.0x0=4.54fwDensity4.55.05.56.06.57.07.50.00.20.40.60.8x0=4.55fwDensity4.55.05.56.06.57.07.50.00.20.40.60.81.0x0=4.56不同原点的平均直方图4.55.05.56.06.57.00.00.20.40.60.81.0Averageshiftedhistogramforfemalelnwagefwdensityabline(h=)6.22059直方图vs核密度估计直方图密度估计的两大局限最优带宽h不易解决原点的影响即使解决了原点问题，直方图仍然有缺点区间内每个点有相同的密度估计的密度函数不连续解决方法：核密度估计没有原点问题最优带宽得到了较好的解决收敛速度更快由直方图到核密度估计直方图核密度}{#1的小区间内的观察值包含落入某个区间长度xn}{#1的小区间内的观察值附近落入区间长度xn核密度估计niiniiihhxxKnhhxxInhhxhxxnhxf1111211]},[{#21)(ˆ核函数K(u)通常取对称单峰的概率密度函数且满足limu→∞K(u)=0核函数核函数K(u)均匀(Uniform)三角(triangle)Epanechnikov二次权重(quartic/biweight)三次权重(triweight)高斯(Gaussian)余弦(cosine)121uI11uIu11432uIu11161522uIu11323532uIu2exp212u12cos4uIu核密度估计的一般形式hKhKxxKnhxxKhnxfhniihniih1111)(ˆ11其中，LWAGE的核密度估计fwDensity4.55.05.56.06.57.07.50.00.20.40.60.81.0KernelDensityWithDifferentBandwidthBW=0.1112BW=0.05BW=1mwDensity5.05.56.06.57.07.58.08.50.00.20.40.60.81.0KernelDensityWithDifferentBandwidthBW=0.0673BW=0.05BW=1h对核密度估计的影响是什么？LWAGE的核密度估计（续）核函数对核密度估计的影响是什么？fwDensity4.55.05.56.06.57.07.50.00.20.40.60.81.0KernelDensityWithDifferentKernelFunctionguassianepanechnikovtriangularrectangularmwDensity5.05.56.06.57.07.58.08.50.00.20.40.60.81.0KernelDensityWithDifferentKernelFunctionguassianepanechnikovtriangularrectangular核密度估计的统计性质可以证明，对于对称核函数，有：dssKKnhnhoxfKnhxfVardssKsKhoKxfhxfBiashh)(1)(1)}(ˆ{)()()()()(2)}(ˆ{2222222222其中，其中，f(t)核密度估计的均方误差2222422422224224222224)()(41)}(ˆ{1)()()(41)}(ˆ{)}(ˆ{1)()(1)()(4)}(ˆ{xfKhKnhxfAMISEnhohoxfKhKnhdxxfMSExfMISEnhohoxfKnhKxfhxfMSEhhhh最优带宽5/15/1222222~)()(nKxfnKhopt光滑参数的确定-plugin如果变量服从正态分布34.1)()()(ˆ06.1)8(3)2(1)()(ˆˆ21,832)}({34.1,ˆmin06.1]25.0[]75.0[]25.0[]75.0[]25.0[]75.0[5/15/155/12222222255225/1nnnnnnoptoptZZZZXXRnnKxfnhdxxfnIQRh光滑参数的确定-Cross-validation对于任意分布niijnjijhniiihhninjjininjtjininjxjihhxfEfhhhhiijijhijijxxKnnxfnxfEestdvvKvuKuKKhxxKKhndttKthxxKhndxhxxKhxxKhndxxfdxxfdxxffdxxfdxxfxffISEhISExxKnnhxxKKhnhCVhh1,11,112112112222))(ˆ(ˆ222)(111)(ˆ1))(ˆ(.)()(111)(ˆ)()}(ˆ{2)(ˆ)}()(ˆ{)ˆ()()1(21)(其中，无关与的期望可由数据计算得到LWAGE的核密度估计（续）fwDensity4.55.05.56.06.57.07.50.00.20.40.60.81.0KernelDensityWithDifferentBandwithsivermanscottcv-gaussiancv-epanechkovmwDensity5.05.56.06.57.07.58.08.50.00.20.40.60.81.0KernelDensityWithDifferentBandwithsivermanscottcv-gaussiancv-epanechkov置信区间nhKxfzxfnhKxfzxfxfnhnhKxfzKxfhxfnhKxfzKxfhxfxfcnhvzbnxfxfvzbnxfPvzbxfxfnvzbPKxfcKxfcNxfxfncnhfhhhhhhxxhxxhxxhxxvbLh/xx222/1222/15/1222/122222/1225/12/15/22/15/22/15/22/12222525/1)(ˆ)(ˆ,)(ˆ)(ˆ)()()()(2)(ˆ,)()()(2)(ˆ)(}{)(ˆ)(}{)(ˆ)}()(ˆ{1)(1,)()(2)}()(ˆ{2