After I solve a quadratic equation which is the derivative of log-likelihood function to find MLE, I get one positive root and a negative root. Which one do I choose? Sample space is between $$0<y<1$$
If your derivative is a quadratic, your log-likelihood is a cubic. Since you find two roots, your cubic has a local maximum and a local minimum. There are three ways to proceed:
- the "zeroeth derivative test": evaluate the log-likelihood at both roots. Since you want the MLE, pick the root that gives a larger value.
- the first derivative test: On the intervals of the sample space partitioned by the roots, evaluate the derivative to see in which intervals the log-likelihood is increasing/decreasing. A maximum occurs when the function turns (reading left-to-right) from increasing to decreasing.
- the second derivative test: differentiate again and plug the roots into this second derivative. When the second derivative is positive, you have a local minimum. When the second derivative is negative you have a local maximum, which is the one you want to keep for your MLE.
Also, don't forget to evaluate the MLE at the endpoints of the sample space. The derivative of the log-likelihood cannot tell you about maxima at the boundaries of the sample spaces (since nothing interesting happens to the derivative at the boundary of the sample space).