Suppose that X1, ... , Xn form a random sample from a distribution for the pdf f(x0) is as follows: 1 f(x|0) = - 1x-01 for 0 < x < 0 2 Also suppose that the value of 0 is unknown (-<< 0). Find the MLE of 0.
The pdf of each observation has the following form:
Therefore, the likelihood function has the form
It can be seen that the MLE of θ must be a value of θ for which and
which maximizes among all such values. Since is a decreasing function of θ, the estimate will be the smallest possible value of θ such that . This value
is , it follows that the MLE of θ is