For the above question we can find the the log likelihood function and differentiate it with the parameter and equate it to zero and solve it.
But if I do that I get the answer as zero.
What is the correct way to solve this, is there any other way to find maximum likelihood estimate for the parameter?
Best Answer
The reason you usually differentiate the log likelihood function is that it's typically the easiest way to do what you're really trying to do, which is to maximize the likelihood function. So, let's go back to the drawing board and just do what we really want to do -- find $\theta$ that maximizes the likelihood function -- but more directly. The likelihood function, evaluated at the sample data, is $$L(\theta) = \underbrace{1}_{0.2} \cdot \underbrace{\frac{1}{2 \theta - 1}}_{1.2} \cdot \underbrace{\frac{1}{2 \theta - 1}}_{1.4} \cdot \underbrace{1}_{0.3} \cdot \underbrace{\frac{1}{2 \theta - 1}}_{0.9} \cdot \underbrace{1}_{0.7} = \left[ \frac{1}{2 \theta - 1} \right]^3$$ if $\color{blue}{\text{$\theta$ is at least as big as $1.4$}}$. (Note that $L(\theta) = 0$ if $\theta < 1.4$.) So, the question is: for what value of $\theta$ is this as large as possible, subject to the $\color{blue}{\text{constraint}}$? (No derivative necessary -- just think, and pay attention to the constraint.)