How to do "partial" survival analysis on randomly censored data? Suppose that we monitor a population of devices over an interval $[a,b]$. Some devices are added before $a$ and some are added during $[a,b]$. Furthermore, some devices fail during $[a,b]$ and some are healthy until time $b$ and their ages are right censored. Roughly, $90\%$ of the population are right censored. We can assume that the devices are added randomly. We know all device ages during $[a,b]$.
Is this an example of a random censoring? I'm using the following type definitions from Wikipedia:
https://en.wikipedia.org/wiki/Censoring_(statistics)#Types
We are interested to perform survival analysis to estimate the failure distribution $f(t)$ of these devices. My intuition is that since many of the samples are right censored, our estimation of the tail of distribution $f(t)$ will be inaccurate and biased. 
Now, suppose that we are only interested in estimating $f(t)$ over the interval $[0,c]$ where $c < b - a$. Is there a way to obtain an unbiased estimate of $f(t)$ over this interval? Also, we have a lot of data so it is okay to only use a fraction of them. 
 A: To answer your first question, it sounds like random censoring since the censoring time, $b$, is constant for everyone, hence statistically independent of true failure times (How $b$ is chosen may be important, but likely not). 
Though you have lots of right-censored data, you may still have information about the tail. If you use a parametric model, this can give you the entire distribution, even from data that can't be larger than $b$. If you use a non-parametric model, then you are limited to inferences up to your max observation time. 
Here's an example. Suppose your data looks like the following, with $a=10, b=20$. 

A red line is the lifetimes of an observed failure, and a blue line is a censoring event. The black lines denote our observation period, $[10, 20]$. 
Below is the Kaplan-Meier curve for the above data:

Our inference is limited by the largest uncensored observation, which is 14 (=19-5). 
We'll switch to a parametric model. Let's try an Weibull distribution. Below we can see that we can extrapolate to beyond our largest observation (but keep in mind, this is walking on thin ice):

Now, reading your second question, I'm a bit confused. I understood $a,b$ to be wall clock times (see my picture above). But you mention you want inference for $[0, c]$, which doesn't make sense to me. ($[a, b]$ can't be relative to subject time, because you mention adding devices in that time interval). 
