統(tǒng)計(jì)學(xué) 課件 第20講 一元線性回歸的參數(shù)估計(jì)

第20講一元線性回歸的參數(shù)估計(jì)課程類別：專業(yè)必修課授課對(duì)象：經(jīng)管類本科生1引例：地下停車位的市場(chǎng)價(jià)格

相關(guān)因素1.根據(jù)成本確定銷售價(jià)格2.根據(jù)市場(chǎng)價(jià)格確定銷售價(jià)格……2引例：地下停車位的市場(chǎng)價(jià)格調(diào)研周邊的其他樓盤編號(hào)車位價(jià)格（萬(wàn)元）

編號(hào)車位價(jià)格（萬(wàn)元）A011.2188.6A151.56316.2A021.09310.3A161.80415.4A031.0417.5A171.85315.3A041.29310.3A181.0917A051.3659.45A191.0637.1A061.66216.4A201.63515.1A071.5213.2A211.2529.4A082.65921.2A221.088.2A091.50512.2A231.3058.6A101.52512A241.22110.2A112.04818.6A250.9347.3A121.88313.8A261.85318.5A132.71224.1A271.2539.2A142.07215.4A280.8848.63一元線性回歸Stanton,J.M.(2001).Galton,Pearson,andthepeas:Abriefhistoryoflinearregressionforstatisticsinstructors.

JournalofStatisticsEducation,

9(3).FrancisGalton(1822

–1911)4

由于x的變化而引起的y的變化其它隨機(jī)因素對(duì)y的影響

一元線性回歸基本假定線性關(guān)系x的非隨機(jī)性ε是同方差的（σ2是常數(shù)）E(ε)=0ε~N(0,σ2)ε相互獨(dú)立

截距斜率（回歸系數(shù)）x=x2時(shí)的E(y2)x=x1時(shí)y的分布x2x1x=x1時(shí)的E(y1)

xyx=x2時(shí)y的分布

5參數(shù)的最小二乘估計(jì)在二維空間中，這樣的直線似乎是無(wú)窮多的，我們能不能找到最好的？

總體回歸參數(shù)：未知樣本

樣本統(tǒng)計(jì)量代替未知參數(shù)得到估計(jì)的回歸方程6參數(shù)的最小二乘估計(jì)xy(xn

,yn)(x1,y1)(x2,y2)(xi,yi)

7參數(shù)的最小二乘估計(jì)思想：因變量的觀察值與估計(jì)值之間的誤差平方和達(dá)到最小.CarlFriedrichGauss(1777–1855)

最小二乘估計(jì)（LeastSquaresEstimation）最小二乘估計(jì)8參數(shù)的最小二乘估計(jì)

9例題

10編號(hào)車位價(jià)格（萬(wàn)元）

編號(hào)車位價(jià)格（萬(wàn)元）A011.2188.6A151.56316.2A021.09310.3A161.80415.4A031.0417.5A171.85315.3A041.29310.3A181.0917A051.3659.45A191.0637.1A061.66216.4A201.63515.1A071.5213.2A211.2529.4A082.65921.2A221.088.2A091.50512.2A231.3058.6A101.52512A241.22110.2A112.04818.6A250.9347.3A121.88313.8A261.85318.5A132.71224.1A271.2539.2A142.07215.4A280.8848.6例題

編號(hào)A011.2188.610.4751.484A021.09310.311.2581.195A031.0417.57.8081.084A041.29310.313.3181.672A051.3659.4512.8991.863A061.66216.427.2572.762A071.5213.220.0642.310A082.65921.256.3717.070A091.50512.218.3612.265A101.5251218.3002.326A112.04818.638.0934.194A121.88313.825.9853.546A132.71224.165.3597.355A142.07215.431.9094.293A151.56316.225.3212.443A161.80415.427.7823.254A171.85315.328.3513.434A181.09177.6371.190A191.0637.17.5471.130A201.63515.124.6892.673A211.2529.411.7691.568A221.088.28.8561.166A231.3058.611.2231.703A241.22110.212.4541.491A250.9347.36.8180.872A261.85318.534.2813.434A271.2539.211.5281.570A280.8848.67.6020.781

11例題

12例題拓展為什么用豎直距離，不用垂直距離呢？xy(xn

,yn)(x1,y1)(x2,y2)(xi,yi)

13小結(jié)14一元線性回歸：基本假定：最小二乘估計(jì)：

6個(gè)假定思考與練習(xí)思考：在回歸參數(shù)的估計(jì)中，能用矩形面積方法來(lái)估計(jì)嗎？練