Find MLE for $\Phi, f(x;\Phi)=\frac{2x}{1-\Phi^{-1}}$

Question

The probability distribution is given as

$$f(x;\Phi) = \begin{cases} \frac{2x}{1-\Phi^{-1}}& \Phi <x <1\\ 0& \text{otherwise} \end{cases}$$

How do I go about finding the MLE? Having difficulty with deriving the likelihood function

$$L(\Phi)= \frac{2x}{1-\Phi^{2}}$$

Thanks in advance!

Answer 1

There are some issues ti clarify ...

• $\Phi$ has a specific meaning in Statistics...better say $\theta$ for tour parameter

• You did not mention the parameter domain.... I checked it for you... to have a noce density $\theta$ must be $\in [0;1)$

• You did not mention the size of your random sample: let's say $n$

Now you can start.

Your model is not regular as the domain depends on $\theta$ so tour likelihood is

$L(\theta)\propto \frac {1}{(1-\theta^2)^n}\mathbb{1}_{[0;min(x))}(\theta)$

So it is strictly increasing and your MLE is

$\hat{\theta}= min(x)$