Determine the maximum likelihood estimates of a and λ when X1, ..., Xn is a sample from the Pareto density function f(x) = λαλχ-(2+1), if x ≥ a 0 ,if x <a I tried to solve it
The Pareto density function:
[ f(x) = \lambda^n \cdot a^n \lambda \cdot x^{-(\lambda+1)} ]
The logarithm of the likelihood function:
[ \log f = n \log \lambda + n\lambda \log a - (\lambda+1) \sum_{i} \log x_i ]
Taking the derivative with respect to λ and setting it equal to zero:
[ \frac{d \log f}{d \lambda} = 0 ]
Solving for λ:
[ \lambda^* = \frac{n}{\sum_{i} \log x_i} - n \log a ]
Taking the derivative with respect to a and setting it equal to zero:
[ \frac{d \log f}{d a} = 0 ]
We have:
[ \frac{\lambda.n}{a} = 0 ]
Now I'm unable to conclude .