R语言第二次作业（OLS最小二乘估计、三种累积概率分布的比较）

1.

####  work1
x<-runif(20,-5,5)
e1<-rnorm(20,0,1)
e2<-rt(20,3)
e3<-rchisq(20,3)
y1 <-2+10*x+abs(x)*e1
y2 <-2+10*x+abs(x)*e2
y3 <-2+10*x+abs(x)*e3

lm.model1<-lm(y1~x)
lm.model2<-lm(y2~x)
lm.model3<-lm(y3~x)

summary(lm.model1)
summary(lm.model2)
summary(lm.model3)

Call:
lm(formula = y1 ~ x)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.0595 -1.1556 -0.4359  0.4008  5.7312 

Coefficients:
            Estimate Std. Error t value
(Intercept)   2.8672     0.5511   5.203
x            10.1605     0.2261  44.940
            Pr(>|t|)    
(Intercept)    6e-05 ***
x             <2e-16 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.463 on 18 degrees of freedom
Multiple R-squared:  0.9912,    Adjusted R-squared:  0.9907 
F-statistic:  2020 on 1 and 18 DF,  p-value: < 2.2e-16

> summary(lm.model2)

Call:
lm(formula = y2 ~ x)

Residuals:
     Min       1Q   Median       3Q      Max 
-12.9291  -2.5543  -1.9397   0.1668  20.9329 

Coefficients:
            Estimate Std. Error t value
(Intercept)   3.9634     1.6763   2.364
x            10.4164     0.6877  15.146
            Pr(>|t|)    
(Intercept)   0.0295 *  
x            1.1e-11 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.492 on 18 degrees of freedom
Multiple R-squared:  0.9272,    Adjusted R-squared:  0.9232 
F-statistic: 229.4 on 1 and 18 DF,  p-value: 1.096e-11

> summary(lm.model3)

Call:
lm(formula = y3 ~ x)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.0275 -3.8411  0.1872  2.0643 10.8477 

Coefficients:
            Estimate Std. Error t value
(Intercept)   7.7555     1.0674   7.266
x             9.0369     0.4379  20.635
            Pr(>|t|)    
(Intercept) 9.39e-07 ***
x           5.61e-14 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.771 on 18 degrees of freedom
Multiple R-squared:  0.9594,    Adjusted R-squared:  0.9572 
F-statistic: 425.8 on 1 and 18 DF,  p-value: 5.609e-14

通过 summary 函数，可以详细地显示 lm.model 对象中的信息，包括∶第一，模型结构;第二，残差的概述;第三，回归系数及其检验;第四，方程整体效果检验与指标。
可以看出，当随机项服从正态分布时，R-Square达到0.99，效果最好，说明模型的线性效果最强。当随机项服从卡方分布时模型效果较好，达到0.959，当随机项服从t分布时，模型效果最差，但也达到了0.927，可以说模型都取得了较好的效果。

2.

请编程求解图中"红虚线，与水平虚线、垂直虚线"各自相交处的横坐标、纵坐标的具体数值。

ts<-1e-5
x<-0
pextreme <- function(x) 1-exp(-exp(x))      

while(pextreme(x)>0.5){
  x<-x-ts}
paste("x = ",x)
paste("y = ",pextreme(0))

[1] "x =  -0.366520000000226"
> paste("y = ",pextreme(0))
[1] "y =  0.632120558828558"