I have this exercise with solution:

Exercise TZ.5 (MLE - Restricted Support) The density of the Pareto distribution is given by 5 .0: 'f > f(sv;aa)= 5 \"3+ 1 Sta 0 else with parameters a and ,6 > 1. Assume that {3' > 1 is given and that X1,X2, . . .,Xn is an i.i.d. sample. Find the maximum likelihood estimator of a. Solution TZ.5 It is easy to see, that f and thus also the likelihood is monotonically increasing in o: and strictly positive. This means we want a: to be as large as possible. If or becomes too large (larger than one of the X1), the likelihood however becomes zero. So the MLE for a is given by a: min X2- i=1,...,n