Answer:
[tex]\hat \theta = max (X_1,...,X_n)[/tex]
Step-by-step explanation:
We assume the following density function:
[tex] f(\theta) = \frac{1}{\theta} , 0 \leq x\leq \theta[/tex]
And 0 for other case, and we are interested in order to find the MLE for [tex]\theta[/tex]
The likehood function would have the following form:
[tex] L(\theta) = \prod_{i=1}^n f(x_i) = \frac{1}{\theta^n},0\leq x_i \leq \theta , i=1,...,n [/tex]
If we want to maximize this we need a value [tex]\theta[/tex] such that [tex] \theta\geq x_i[/tex] for [tex] i =1,....,n[/tex]
Our likehood function is a decreasing function and in order to estimate this value we can use the maximum function of all the observations:
[tex] \theta = max (x_1,....,x_n)[/tex]
And the the MLE estimator for the parameter [tex]\theta[/tex] is given by [tex]\hat \theta = max (X_1,...,X_n)[/tex]
Is important to mention that this estimator probably underestimate the value of [tex]\theta[/tex] since [tex] max (X_1,...,X_n)< \theta[/tex], but is the stimator for this case.
