Consider the independent non-identical random variables $X_i \sim Exp(i/\mu)$ , enumerated by $i \in \mathbb{N}$. Here, $\mu > 0$ is an unknown parameter. Instead of the entire sample $x_1, . . . , x_n$, suppose you are only given the sample minimum, $x_{(1)} = min\{x_1, . . . , x_n\}$. Based on the distribution of the sample minimum $X_{(1)}$, which you should derive, find the maximum likelihood estimate $\mu(min)$ of $\mu$ in this case.
--
My initial approach was to find the CDF of the minimum and differentiate to achieve the PDF of the minimum. This PDF has a product function instead of nth power since they are non-identical. To my understanding, this PDF is now the likelihood function, and I continued to solve the problem like any other MLE problem i.e. take logs, differentiate, set to 0 and find the mu min. However, my result achieves the same result as solving the MLE problem with a whole sample distribution. I feel that is incorrect and so wondering, if anyone can provide a model solution.
thanks