Estimating target variable without data I have a metric labor hrs per unit, $L$, that is currently tracked by:$$\frac{\text{clocked worker hours}}{\text{total output}} $$
on a per week basis. Unfortunately, it does not matter what type of unit is being worked on, the labor hrs for a specific type of unit is not accounted for in the denominator.  There is a particular unit denoted by $U$, that I need to estimate $L$ for. Hence, I think I might want to estimate something like: $$L_U = \frac{\text{clocked worker hours}}{\text{total output for $U$}}$$
There is no data available for something like this and all I have is the amount $L$  per week and the count of $U$ that comes in every week (the incoming $U$ does not mean that all of it is going to be part of the total output for $U$, some of those incoming units get scrapped and do not count in the overall output).
The only thing I can think of is to collect live samples from a time study and use Kernel Density Estimation (KDE) to estimate the distribution of $L_U$ (after I have figured out what features may effect this from the domain experts). Once the distribution is estimated, I can randomly sample "synthetic" data from it and use it to estimate $L_U$. I am also going to try to do some rule-based modeling based on the estimates/intuition of domain experts but I feel like that might be a long shot.
Question 1:
Would like to know how others would tackle this problem (preferably without a time study/collecting live samples) and if my current approach makes sense to even attempt?
Question 2:
If I were to be able to estimate this with the sheer lack of data at my disposal, how would I provide a measure of confidence/certainty? Surely, there would be a massive lack of statistical integrity?

EDIT: I was able to get access to real-life samples, but $< 10$. I have the current framework set up for this project of mine, and other particular questions:
Assume you have $< 10$ real-life observations of a variable X.

*

*Step 1: Use Kernel Density Estimation on the $< 10$ samples to
approximate the distribution for $X$.


*Step 2: Randomly sample 10,000 observations from the resulting
distribution from Step 1 therefore creating synthetic data.


*Step 3: Iterate on Step 2 using MC simulation to extract estimates
for the population mean and variance of $X$.
Questions:

*

*Question 1: Does this approach seem robust? Should I verify it with a hypothesis test by gathering a decent number of real samples once I am able to? What other measures of confidence can I use?


*Question 2: Should the KDE step be incorporated into the MC
simulation? If so, how/why?


*Question 3: Does my choice of kernel matter? If so, how do I choose the appropriate type of kernel?
 A: < 10 is probably not enough samples for KDE. See the answer to a specific question about KDE and sample size  (reference given in the answer).
If you have $U$ and worker hours (the numerator of $L$), you can actually use simple linear regression to infer worker hour per unit of $U$ produced.
Let clocked worker hours =$C$
$C= \beta_0 + \beta_1*U +e$
The estimate for $\beta_1$ is the change in worker hours for a one-unit increase in $U$. This assumes a linear relationship, that there are no confounders (i.e. there's no underlying process like "When we increase production of $U$, we always increase this other labor-intensive product $V$."), and that your values are independent.
You can use your actual data as a hold out sample to see if it matches the relationship you've drawn. In my opinion, that small sample of real data is most valuable as a test set for estimates you are getting with the larger data.
A: Before going to machine learning, I think it will clarify your approach to the problem to formulate it first as a linear regression.  After you have identified a regression that clearly describes the relationship between the variables that is physically meaningful and compatible with your data set, then consider more elaborate regressions.
For example, if you had columns for worker (categorical), unit processed (categorical), hours labored on the unit (numeric), you could regress hours ~ worker + unit.  Worker and unit might be treated as random effects if you have a large number of levels.  The coefficients will indicate the hours required to labor on each unit, and whether a worker has more or fewer hours on average.  A regression with interaction terms hours ~ worker*unit can also describe different labor times for each worker on each type of unit (if worker B is fast at processing unit delta  but slow at processing unit gamma).
If you only had the total time per week, you might instead create the model total_time ~ unit_alpha + unit_gamma + worker ... where unit_alpha represents the number of alpha units that a worker processed during that time.  Again, the coefficients for worker will indicate if a worker took more or less time than other workers, and coefficients for units will indicate how long each unit takes to process.  You can also consider terms for worker-unit interactions, random effects treatment of worker, and estimated marginal means for more rigorously measuring "the processing time for unit alpha for a typical worker and its standard error."
