Simulation study for a Exponential Distribution

Question

I have a pdf defined as: $$f(x)=\lambda e^{-\lambda(x-b)}$$

Conduct a simulation study in R to explore the behaviour of the maximum likelihood estimator $λ_{MLE}$ for λ on simulated data $X_1, · · · , X_n$ (as independent copies of X with parameter λ) according to the following instructions. Take b = 0.01 and consider a setting in which λ = 2 and generate a plot of the mean squared error as a function of the sample size n. You should consider a sample size between 100 and 5000 in increments of 10, and consider 100 trials per sample size. For each trial of each sample size generate a random sample $X_1, · · · , X_n$ (as independent copies of X with parameter λ = 2), then compute the maximum likelihood estimate $λ_{MLE}$ for λ based upon the corresponding sample. Display a plot of the mean square error of $λ_{MLE}$ as an estimator for λ as a function of the sample size n.

I don't really understand how to go about the simulation study. I don't need an answer, I just need an outline on how to approach the simulation. Probably should use a dexp() function in R right? What do they mean by a 100 trials in each sample size. Please help!

Answer 1

Since you do not want a complete answer and just an outline to follow, you can proceed as below:

1. Define a function lambda_hat(sample) which takes a sample and computes its MLE for the parameter $\lambda$.
2. Define another function to compute the Mean Square Error (MSE) of the estimator, as below:

MSE <- function(sample_size, lambda = 2, size = 100) {
    output <- 0
    # write a loop and in each iteration take a sample by rexp() function
    # compute its estimate for lambda and add the square error to `output`
    # at the end return `output` divided by `size`
    return(output/size)
}

Note that to compute the MSE of an estimator you need to compute the estimate a number of times and then take its average, which the instructions wants you to do that 100 times, and each time to compute the estimate you need a sample of a fixed size which comes from the vector x below.

3. Define a sequence x <- seq(100, 5000, by = 10).
4. Compute the mean square error for each sample size in x by y <- sapply(x, MSE).
5. Plot by plot(x, y).