I want to calculate the parameter $\lambda$ of the exponential distribution $e^{-\lambda x}$ from a sample population taken out of this distribution under biased conditions. As far as I know, for a sample of n values, the usual estimator is $\hat{\lambda} = \frac{n}{\sum x_i}$. However my sample is biased as follows:
From a complete population of m elements drawn i.i.d from the exponential distribution, only the n smallest elements are known. How can I estimate the parameter $\lambda$ in this scenario?
A bit more formaly, if $\{x_1,x_2,x_3,...,x_m \}$ are iid samples drawn from $e^{-\lambda x}$, such that for every $i < j$ we have $x_i \leq x_j$, then how can I estimate $\lambda$ from the set $\{x_1,x_2,x_3,...,x_n\}$ where $n < m$.
Thanks a lot!
Michael