Parameters estimations with right censored data As I was doing some research about survival analysis, I found the following article stating that, if we have right censored data, we can estimate the parameters of a distribution with the following modified MLE :
$L(\theta, X) = \prod_{i=1}^{n}f(x_i;\theta)^{\delta_i} [1-F(x_i;\theta)]^{1-\delta_i}$
$\delta_i = \begin{cases} 1 \quad \text{if $x_i$ is censored} \\ 0 \quad \text{if $x_i$ is not censored} \end{cases}$
My question would be the following : If, in some cases, we only have access to right censored data, would it still possible to have a estimation of the parameters using :
$L(\theta,X) = \prod_{i=1}^{n}(1-F(x_i; \theta))$
If you have additional content on the topic, I would be very happy to have a look on it.

No, it is not a problem!  For a post with an example see

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*ML estimate of exponential distribution (with censored data)
Note that for the exponential distribution example there, when all observations are censored, it is reduced to a binomial likelihood where the probability parameter is a function of the rate $\lambda$.
For a good reference treating this, see In All Likelihood: Statistical Modelling and Inference Using Likelihood
by Yudi Pawitan (Chapter 11).
