Suppose that random variables $X_i, \ldots, X_n$ are independent and identically distributed with the CDF:
$$F(x|a,b) = 1-\left(\frac{b}{x}\right)^a$$
With the conditions $x \ge b, b \gt 0$ and $a \gt 0$. And for $x \lt b$, the distribution is $0$.
What are the maximum likelihood estimators for $a$ and $b$?
I approached this by taking the derivative of $F(x)$ to get the PDF, $f(x)$. Then, I did the product from $i=1$ to $i=n$ of $f(x_i)$ to get the likelihood functoin. I took the log of the likelihood, then partial derivatives with respect to $a$ and $b$, set the partials to $0$, and ended up with this MLE for $a$:
$$a = \frac{n}{\sum_{i=0}^n (x_i) - n\ln(b)}$$
For $b$, I got this as the partial derivative w/respect to $b$:
$$\frac{an}{b}$$
So if we set that to $0$, there is no solution and I can't get the MLE of $b$. Where do I go from here?
Thank you!