Maximum Likelihood of single observation

Question

I'm stumped on this problem... I keep getting an undefined answer of having to solve -20 = 0. The likelihood function I get is $e^{-20\alpha}$.

So I have

• $y_i=$ $ \begin{cases} 1& w/probability \ p_i \\ 0& w/probability \ 1-p_i \end{cases}$

• And $p_i=1-e^{-\alpha x_i}$, i=1,2,...

• $\alpha>=0$ and $x_i>=0$

If I have a sample of a single observation $(y_1,x_1)=(0,20)$, I am looking for the maximum likelihood of this sample.

Thank you for any help :)

Answer 1

Hint:

• You have a likelihood of $L(\alpha\mid y_1=0) = e^{-20\alpha}$ and you also know $\alpha\ge 0$

• If $e^{-20\alpha}$, which is continuous and differentiable, does not have a turning point for positive $\alpha$, then it is monotonic and will have a maximum either when $\alpha=0$ or when $\alpha \to \infty$, and it is fairly obvious which