In class we derived the MOM and MLE for an exponential distribution with parameter ????. Conduct a Bootstrap simulation to compare the estimation of λ with sample sizes of n = 10, n = 100, and n = 500. Choose true value λ = 0.2 and use B = 1000. Calculate and compare the mean and standard error for each set of simulations to each other as well as their theoretical values.

To conduct a Bootstrap simulation to compare the estimation of the parameter λ of an exponential distribution with different sample sizes, let's follow these steps:

1. Generate a sample size of `n` from the exponential distribution with the true parameter λ = 0.2. Do this for n = 10, n = 100, and n = 500. For each sample, calculate the Method of Moments (MOM) and Maximum Likelihood Estimation (MLE) for λ.

Note: For an exponential distribution, the MOM estimator is the sample mean (1 / sample mean), and the MLE estimator is also the sample mean (1 / sample mean).

2. Perform B = 1000 Bootstrap resamplings from each original sample. For each resample, calculate the MOM and MLE estimates. This will give you 1000 estimates of λ for each sample size and each method.

3. Calculate the mean and standard error of the 1000 Bootstrap estimates of λ for each sample size and each method. This will provide you with the Bootstrap mean and standard error for MOM and MLE for each sample size.

4. Compare these Bootstrap mean and standard error values for each sample size to each other, as well as to their theoretical values. The theoretical mean of the estimator for λ should be λ itself, which is 0.2. The theoretical standard error for the MLE of an exponential distribution is λ / sqrt(n).

Now, let's look at the results you might expect:

For n = 10, you will likely see larger variability in the Bootstrap estimates with a higher standard error compared to larger sample sizes. The mean estimate for λ might also be less accurate.

For n = 100, the Bootstrap estimates should have a lower standard error and the mean estimate should be closer to the true value of λ.

For n = 500, the estimates should be even more accurate and less variable, with the mean very close to the true value of λ and the standard error should be smaller, reflecting the increased precision due to the larger sample size.