Question

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.

Answer 1

The pdf of each observation has the following form:

$f(x|θ) = (\dfrac{1}{θ}, for 0 ≤ x ≤ θ$

      $(0, otherwise$

Therefore, the likelihood function has the form

$L(θ) = ( \dfrac{1}{θ^n} , \text{ for } 0 ≤ x_i ≤ θ (i = 1, · · · , n)$

      $( 0 , otherwise$

It can be seen that the MLE of θ must be a value of θ for which $θ ≥ x_i, \text{ for } i = 1, · · · , n$ and

which maximizes $\dfrac{1}{θ^n}$ among all such values. Since $\dfrac{1}{θ^n}$ is a decreasing function of θ, the estimate will be the smallest possible value of θ such that $θ ≥ x_i \text{ for } i = 1, · · · , n$ . This value

is $θ = \text{ max }(x_1, · · · , x_n)$ , it follows that the MLE of θ is $\hat{θ} = max(X_1, · · · , X_n)$