4
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.