Let X1, . . . , Xn, . . . be iid random variables with common PDF f(x | θ) = e^−(x−θ) x > θ, 0 x ≤ θ. We wish to show that the MLE ^θn = X(1) is consistent.   (a) Why can you not directly apply the MLE theorem to obtain this conclusion? (b) Prove that ^θn is consistent by using the CDF for X(n) and the definition of convergence in probability. (c) Prove that ^θn is consistent by considering the bias and variance of the estimator.

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