true_theta <- 3.14 # true value of parameter
n <- 25 # sample size
M <- 2500 # how many repetitions
estimates <- numeric(M) # preallocate storage for the sample of estimates
for(m in 1:M){
# Step 1: simulate a new fake dataset of size n
# Step 2: apply your formula from Task 2 to compute new estimate
# Step 3: store it
}
# Step 4: plot histogram of estimates
# Step 5: add density curve of exact sampling distributionLab 10
Due Thursday November 20 at 11:59 PM
This lab provides extra practice with the kind of maximum likehood problem that you will encounter on the last problem set and the final exam.
Task 1
If a non-negative random variable \(X\) has the Rayleigh distribution with parameter \(\theta>0\), then its CDF is
\[ F(x;\,\theta)=1-\exp\left(-\frac{x^2}{2\theta}\right)\quad x\geq 0, \]
and we say \(X\sim\text{Rayleigh}(\theta)\). What is the PDF in this parametric family?
Task 2
Imagine we observe data
\[ X_1,\,X_2,\,...,\,X_n\overset{\text{iid}}{\sim}\text{Rayleigh}(\theta), \]
where the true value of the parameter \(\theta>0\) is unknown. Use the method of maximum likelihood to propose an estimator for \(\theta\).
Task 3
You estimator in the previous part is a random variable that inherits its randomness from the \(X_i\). Derive the full sampling distribution of your estimator.
Task 4
Now that you know the sampling distribution of the estimator, compute its mean squared error (MSE) using the bias-variance decomposition. Is the estimator unbiased? Is it consistent?
Task 5
Compute the quantile function of the Rayleigh distribution and use it to generate 100 draws from the distribution that has \(\theta=4\).
Task 6
Write a for loop that simulates the sampling distribution for your estimator. Here is a schematic of what you are doing:
\[ \begin{matrix} \text{0. True parameter value} &&& \theta_0 && \\ &&& \downarrow && \\ \text{1. True distribution} &&& \text{Rayleigh}(\theta_0) && \\ &\swarrow &\swarrow& \cdots &\searrow&\searrow \\ \text{3. Simulated data}&x_{1:n}^{(1)} &x_{1:n}^{(2)}& \cdots &x_{1:n}^{(M-1)}&x_{1:n}^{(M)} \\ &\downarrow &\downarrow& \cdots &\downarrow&\downarrow \\ \text{4. Different estimates}&\hat{\theta}_n^{(1)} &\hat{\theta}_n^{(2)}& \cdots &\hat{\theta}_n^{(M-1)}&\hat{\theta}_n^{(M)} \\ &\searrow &\searrow& \cdots &\swarrow&\swarrow \\ & && \text{Histogram} && \\ \end{matrix} \]
At the end, add on top of the histogram a curve of the density that you derived in Task 3. They ought to match.
Template
Here is some template code you can fill in: