ST117 E3

This submission was created by:

1. Name and WARWICK ID: Questions a1, a2,…
2. Name and WARWICK ID: Questions b1, b2,…
3. Name and WARWICK ID: Questions c1, c2,…
4. Name and WARWICK ID: Questions d1, d2,…

• Please read the document “Guidance on Exercise Sets” on Moodle. Remember that from E2 onwards 10% of the total credit is dedicated to style.
• Please replace xyz and X in the header by the correct codes for your Homework Pod.
• Please list all Homework Pod members who contributed to these solutions and which questions/parts they are responsible for (final write up).
• Remember that you are responsible for everything that you submit even if you used any tools or resources to help you complete your coursework and that you expected to be able to walk others through your calculations, reasoning, and R code explaining the rationale for each step.
• Your course handbooks contain “AI Guidance for Statistics modules” in Section 5.3., and policy about academic integrity in Section 5.4. Course handbooks are available from Moodle (Statistics > ST_Handbooks).

Question A

1.

set.seed(3)
lambda <- 2
vartheta <- 1 / lambda  
n <- 5
sim_num <- 12345      


samples <- matrix(rexp(n * sim_num, rate = lambda), nrow = sim_num)


vartheta1_hat <- rowMeans(samples)           
vartheta2_hat <- rowSums(samples) / (n + 1)   


bias1 <- mean(vartheta1_hat) - vartheta
bias2 <- mean(vartheta2_hat) - vartheta
var1 <- var(vartheta1_hat)
var2 <- var(vartheta2_hat)
mse1 <- mean((vartheta1_hat - vartheta)^2)
mse2 <- mean((vartheta2_hat - vartheta)^2)

cat("Theoretical Results (n=5, λ=2):\n",
    "ϑ̂₁: Bias = 0, Variance =", vartheta^2 / n, ", MSE =", vartheta^2 / n, "\n",
    "ϑ̂₂: Bias =", -vartheta / (n + 1), ", Variance =", (n * vartheta^2) / (n + 1)^2, ", MSE =", vartheta^2 / (n + 1), "\n\n")

## Theoretical Results (n=5, λ=2):
##  ϑ̂₁: Bias = 0, Variance = 0.05 , MSE = 0.05 
##  ϑ̂₂: Bias = -0.08333333 , Variance = 0.03472222 , MSE = 0.04166667

cat("Simulation Results:\n",
    "ϑ̂₁: Bias =", bias1, ", Variance =", var1, ", MSE =", mse1, "\n",
    "ϑ̂₂: Bias =", bias2, ", Variance =", var2, ", MSE =", mse2, "\n")

## Simulation Results:
##  ϑ̂₁: Bias = -0.001185451 , Variance = 0.05075666 , MSE = 0.05075396 
##  ϑ̂₂: Bias = -0.08432121 , Variance = 0.03524768 , MSE = 0.04235489

2.

#Advantages**: Unbiased. On average, it equals the true value \(\vartheta\).  
#Disadvantages**: Larger variance, especially for small samples, leading to higher uncertainty.  
#Advantages**: Smaller MSE, meaning it has lower overall error compared to \(\hat{\vartheta}_1\
#Disadvantages**: Introduces a negative bias, systematically underestimating \(\vartheta\).  

Question B

1.

2.

3.

Question C

1.

2.

3.

4.

5.

Typically, solutions to an exercise contain the following components:

Some text explaining how you approach the task…

Theoretical calculations (if needed), including assumptions and rationales

# definitions of functions
# commented R commands 

Figures (if applicable)

Some text explaining what has been achieved, interpretations, and answers to the questions in the description of the task.