Suppose I have a variable Y that I want to predict with a model using predictor variables X1, X2 and X3. I have a large set of Y-data and from this I know with some certainty and accuracy the empirical distribution of Y.

Assume I have no data where X1...X3 is paired with Y. However, I know from domain knowledge that Y is (say) directly related to X1 and X2, and indirectly related to X3.

Now, given a new data set of say 1000 observations where each observation has values for X1, X2 and X3, I want to predict Y for each observation based on my domain knowledge such that the resulting distribution of predicted Y-values matches the known empirical distribution of Y.

Thus to summarize: I have no data on which to train a model, but I do have domain knowledge. My main interest is to get - for a set of predictor variables - a predicted Y-distribution that matches the previously observed Y-distribution.

What tools and methods can I use to approach this problem? Many thanks!


closed as off-topic by user158565, Michael Chernick, Peter Flom - Reinstate Monica May 19 at 13:22

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  • $\begingroup$ Without data on which to train a model, I don't think this is a statistics problem. $\endgroup$ – Peter Flom - Reinstate Monica May 19 at 13:22
  • $\begingroup$ For anyone who comes across this question - I have found the following paper very useful: "Model Calibration Under Uncertainty - Matching Distribution Information" by Swiler, Aldred and Adams. link. What these authors look at is how to calibrate a model to match a target outcome distribution, rather than a mean predicted value. This is useful if you need a predictive model to simulate a distribution in one of the predictor variables. $\endgroup$ – Fritz45 Jul 11 at 22:25