I need someone's insight on applying a MLE for an exponential distribution. In a finance paper, I have the following:
$\displaystyle d_i \sim \frac{\epsilon_i}{\lambda_i}$ where $\epsilon_i$ is i.i.d. exponentially distributed with parameter $= 1$.
and $i=1,\ldots,n$.
$d_i$ are duration time values like time between two events. The $\epsilon$ are not observed. $\lambda_i$ are not observed and must be replaced with estimates from an optimal filter under a $2^k$ states where $k$ can take value $2 \ldots 10$.
Conditional on $\lambda_i$ the $d_i$ have an exponential distribution of $\lambda_i$ with density $p(d_i|\lambda_i) = \lambda_i \exp[-\lambda_i d_i]$
The $\epsilon_i$ in $\displaystyle d_i \sim \frac{\epsilon_i}{\lambda_i}$ confuses me in the MLE application. First, is the $\epsilon_i$ relevant in the MLE computation? If yes, how does it influence the likelihood fucntion below:
$ \mathcal{L}(\lambda,d_1,\dots,d_n)=\prod_{i=1}^n f(d_i,\lambda)=\prod_{i=1}^n \lambda e^{-\lambda d}=\lambda^ne^{-\lambda\sum_{i=1}^nd_i} $