Assume a continuous time Markov chain which is run through in one direction and finally absorbed at the last state
$1 \rightarrow 2 \rightarrow 3 \rightarrow ... \rightarrow n $
The transition matrix is
$\begin{bmatrix} -\lambda_1 & \lambda_1 & 0 & 0 & 0\\ 0 & -\lambda_2 & \lambda_2 & 0 & 0\\ 0 & 0 & ... & ... & 0\\ 0 & 0 & 0 & -\lambda_{n-1} & \lambda_{n-1}\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$
Now suppose we have different realizations $i$ of this chain. For each realization we know the initial state $a_i$, and the state $b_i$ at some (known) time $t_i$ afterwards.
Is there an efficient way to estimate the mean total transition time
$\sum_{j=1}^{n-1} \frac{1}{\lambda_j}$
from this data?