Help needed for c (iii)

Let X ~ Unif(0, 0), a uniform distribution with an unknown endpoint 0. (a) Suppose we have a single observation on X. i. Find the method of moments estimator (MME) for 0 and derive its mean and variance. ii. Find the maximum likelihood estimator (MLE) for e and derive its mean and variance. (b) The mean square error (MSE) of an estimator is defined as MSE(0) = E (0 - 0). i. Let bias(0) = E(0) - 0. Show that, MSE(0) = var(0) + bias(0)2. ii. Compare the MME and MLE from above in terms of their mean square errors. iii. Find an estimator with smaller MSE than either of the above estimators. (c) Suppose we have a random sample of size n from X. i. Find the MME and derive its mean, variance and MSE. ii. Find the MLE and derive its mean, variance and MSE. iii. Consider the estimator ad where 0 is the MLE. Find a that minimises the MSE. Some information that might be useful: E (X(1) = E (X 1)) = 202 no2 n+1 (n + 1) (n + 2) ' EX(n)= ne n+1 EX(n) = n + 2