Answer to Question #178317 in Statistics and Probability for Priyanka Sarkar

Question #178317

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.

Expert's answer

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

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