线性回归模拟

Question

模拟线性回归的条件，并表明多维线性回归（三个或更多参数）的估计是无偏的。尝试对线性回归的参数做出有偏差的估计，并通过模拟显示您已成功实现了偏差。

这是我尝试过的方法，但我始终坚持从有偏估计中获得无偏估计。

b0=2
b1=1.3
b2=5
N=1000
matrica=matrix(rep(0,N*3),ncol=3)
for (i in 1:N) {
      x1=rnorm(100) ##expectation and       variance is arbitrary
      x2=rnorm(100)
      err=rnorm(100)
      y=b0+b1*x1+b2*x2+err
     lm=lm(y~x1+x2)
     matrica[i,1]=lm$coefficients[1]
     matrica[i,2]=lm$coefficients[2]
     matrica[i,3]=lm$coefficients[3]
    }
   matrica


   rez1 <- matrica[1:N ,1]
   rez2 <- matrica[1:N ,2]
   rez3 <- matrica[1:N ,3]

   ## now we need to show that the      estimates are unbiased     (b0~mean(rez1...))
   summary(rez1)
   summary(rez2)
   summary(rez3) 

  cor(rez1,rez2)  #highly connected 
  cor(rez2,rez3)  #highly connected

Answer 1

与开始的方式非常相似，您可以执行以下操作：

# True Values
b0=2
b1=1.3
b2=5

# Simulation Set Points
N=1000
n <- 100
set.seed(42)

collector <- matrix(ncol = 3,nrow = N)
colnames(collector) <- c("b0_hat", "b1_hat", "b2_hat")
for(i in 1:N){

  # Generate Data
  x1 <- rnorm(n, mean = 1, sd = 1)
  x2 <- rnorm(n, mean = 1, sd = 1)
  y_hat <- b1 * x1 + b2 * x2 + b0

  # Add Noise
  y <- rnorm(n, y_hat, 1)

  # Fit Data
  fit <- lm(y ~ x1 + x2)

  # Store Results
  collector[i, ] <- fit$coefficients
}

然后要显示结果，您可以绘制直方图并显示估计的平均值接近真实参数值beta。

# Graph to Show UnbiasedNess
par(mfrow = c(3,1))
hist(collector[,1], main =expression(hat(beta[0])),breaks = 30)
abline(v =b0, col = "red", lwd = 2)

hist(collector[,2], main =expression(hat(beta[1])),breaks = 30)
abline(v =b1, col = "red", lwd = 2)

hist(collector[,3], main =expression(hat(beta[2])),breaks = 30)
abline(v =b2, col = "red", lwd = 2)

有偏估计是指从长远来看（即期望值），参数估计与真实参数值不同。一种实现方法是说，错误不是从正态分布中提取，而是从t分布中提取。

# True Values
b0=2
b1=1.3
b2=5

# Simulation Set Points
N=1000
n <- 100
set.seed(42)

collector <- matrix(ncol = 3,nrow = N)
colnames(collector) <- c("b0_hat", "b1_hat", "b2_hat")
for(i in 1:N){

  # Generate Data
  x1 <- rnorm(n, mean = 1, sd = 1)
  x2 <- rnorm(n, mean = 1, sd = 1)
  y_hat <- b1 * x1 + b2 * x2 + b0

  # Add Noise from a t-distribution using `rt`
  y <- rt(n, df = 3, ncp = y_hat)

  # Fit Data
  fit <- lm(y ~ x1 + x2)

  # Store Results
  collector[i, ] <- fit$coefficients
}

现在您可以在下面的代码中看到我们的估算值存在偏差。

# Graph to Show UnbiasedNess
par(mfrow = c(3,1))
hist(collector[,1], main =expression(hat(beta[0])),breaks = 30)
abline(v =b0, col = "red", lwd = 2)

hist(collector[,2], main =expression(hat(beta[1])),breaks = 30)
abline(v =b1, col = "red", lwd = 2)

hist(collector[,3], main =expression(hat(beta[2])),breaks = 30)
abline(v =b2, col = "red", lwd = 2)