Suppose I have data $x$ and $y$, where $x$ is a count and $y$ is continuous. I would like to predict $x$ from $y$.
Specifically, for my research question, $X$ can be viewed as being measured without error (it is fixed by design), whereas $Y$ is random.
Below is my data
$x$:  1 2 3 4 5 6 7 8 9 10 $y$:  1.0000 1.8002 2.4383 2.9353 3.3641 3.6847 3.9578 4.1610 4.3139 4.4667
The above data come from a simulation that I have developed to generate asymptotic "accumulation" curves. Basically, my simulation randomly samples without replacement from a pool of distinct character labels and computes the mean across all individuals (represented by the $x$ data). For the above data there are 5 character labels. I want to see if I can recover all 5 distinct labels. Based on the above data, only 4.4667 labels have been recovered on average.
What I am looking for is a regression technique that I can apply to the kind of data that I have supplied.
Specifically, I would use the proposed regression method to answer a question such as "What is the value of $x$ for a corresponding $y$-value of $y$ = 5?" That is, in the context of my data, what $x$ is needed to observe exactly $y$ = 5 labels?
I am unaware of existing appropriate alternatives that could work in this setting besides inverse regression.