How to estimate parameters of a distribution with left-truncated and right-censored data? I have been trying to find the best way of estimating parameters for a known pdf from a data-set that is left-truncated and right-censored.
More precisely, I have lifetimes for a system where there are no logs kept for values under $t_L$; and there are censored values on the right side, where the parts are taken out of the system at a certain time which all the values are greater than $t_R$.
I know how to formulate MLE for censored values, however, I am having problems doing the same thing with the truncated part.  
 A: I found an answer to this question and I am posting it here for the community benefit.
In the case of left-truncated data, the likelihood function can be written by normalizing the truncated pdf. Let $f(x|\theta)$ to be the probability distribution function, $F(x|\theta)$ to be the cumulative distribution function, and $S(x|\theta) = 1 - F(x|\theta)$ to be the survival function.
If the data is unavailable for $x<a$, such as the data is left-truncated at $a$, we can rewrite the pdf and the survival function:
$$\tilde{f}(x|\theta) = \frac{f(x|\theta)}{S(a|\theta)},$$
$$\tilde{S}(x|\theta) = \frac{S(x|\theta)}{S(a|\theta)}.$$
Now, using these new functions, the likelihood for the right-censored data is
$$\mathcal{L}(\theta\,;\,x_1,\ldots,x_n, x_{n+1}, \ldots, x_m) = \prod_{i=1}^n \tilde{f}(x_i\mid\theta) \prod_{i=n+1}^m \tilde{S}(x_i\mid\theta),$$
where $x_1,\ldots,x_n$ are uncensored data points and $x_{n+1}, \ldots, x_m$ are right-censored data points.
For more information, Klein JP, Moeschberger ML. Survival Analysis, Techniques for Censored and Truncated Data. Springer Science & Business Media; 2005, Chapter 3.
