Question

SOLVE the following in R code:

iid Let X1, , Xn ~ U (0,0). We are going to compare two estimators for θ: 01-2X, the method of moments estimator -maxX.... X1, the maximum likelihood estimator I. Generate 50,000 samples of size n-50 from U(0,5). For each sample compute both θ1 and 02 (Hint: You can use the R cornmand max (v) to find the maximum entry of a vector v). The results should be collected in two vectors of length 50,000 2. Plot two histograms: one for the vector of θ's, and another for the vector of 0's. 3. For both the vector of 's and the vector of2's,estimate the (a) Bias (Hint: this can be computed by taking X-5.) (b) Variance (c) Mean Squared Error (Hin this can be computed by adding the variance, and the squared bias. 4. Between 01 and 02, which should be preferred on the basis of the MSE criteria?

Answer 1

# Answer to question 1: generating 50,000 samples of size 50 from U(0,5)

rm(list=ls())
df <- data.frame(matrix(ncol = 2, nrow = 0))
colnames(df) <- c("theta1", "theta2")
for(i in 1:50000){
df1 <- data.frame(matrix(ncol = 2, nrow = 1))
colnames(df1) <- c("theta1", "theta2")
X <- runif(50,0,5)
df1["theta1"] <- 2*mean(X)
df1["theta2"] <- max(X)
print(df1)
df <- rbind(df,df1)
}

theta1 <- df$theta1
theta2 <- df$theta2

# Answer to question 2: ploting histogram og theta1 and theta2

hist(theta1)
hist(theta2)

# Answer to question 3: Calculating Bias and Variance for theta1 and theta2
# Answer 3 (a)
Bias_theta1 <- mean(theta1)-5
Bias_theta2 <- mean(theta2)-5
# Answer 3 (b)
Variance_theta1 <- var(theta1)
Variance_theta2 <- var(theta2)
# Answer 3 (c)
MSE_theta1 <- Bias_theta1^2 + Variance_theta1
MSE_theta2 <- Bias_theta2^2 + Variance_theta2
  

# Answer to question 4 : theta2 should be preffered based on MSE