Let's say that I have a bunch of independent samples, $X_1, X_2, \dots, X_n$ and that they all follow Exponential($\theta_i$) distributions. (So they all have pdf $f(x_i)=\theta_i\exp(-\theta_iy_i)$.) I don't know if all the $\theta_i$s are equal or not, so I will assume the worst and say they are not for generalization purposes. How do I find the maximum likelihood estimate of this?
Here's my work so far:
$L = L(\theta_1, \theta_2, \dots, \theta_n | x_1, x_2, \dots, x_n)=\prod \theta_i \exp(-\sum \theta_iy_i)$
$\ln(L)=\sum\ln(\theta_i) – \sum\theta_ix_i$
$d/d\theta_i=\sum\frac{1}{\theta_i} – \sum x_i = 0$
$\sum \theta_i = \sum \frac{1}{x_i}$
Here's where I'm stuck – how can I say anything about a single $\theta_i$? Is $\hat{\theta}_i=\frac{1}{x_i}$??
Best Answer
$\ln(L)=\sum\ln(\theta_i) - \sum\theta_ix_i$
$\partial/\partial\theta_i=\frac{1}{\theta_i} - x_i = 0 $ $\implies$ $\hat\theta_i = \frac{1}{x_i}$
And $\partial^2/\partial^2\theta_i=\frac{-1}{\theta_i^2} < 0\ \forall\ \theta_i$ so $\hat\theta_i=\frac{1}{x_i}$ is indeed where the maxima occurs.