Regression with distribution as y variable rather than a single value?

I am interested in examining the relationship of X on Y. Both X and Y are continuous variables. I expect the relationship is linear. I know I can do a simple linear regression here, however I'm curious if I could do something a bit different to make a more informed inference.

The Y values are themselves estimated parameters from another test. I have an estimated mean and standard deviation for each Y value (from normal distributions). Could I use the whole probability distribution from each Y observation rather than just the mean estimate when examining Y's relationship with X? The standard deviations for different Y values can vary quite a bit, so I think it could be interesting to see how their distributions may overlap and combine and whether that will affect the statistical relationship between X and Y.

Does anyone know what kind of test I can do to examine this? I'm analyzing my data in R.

Here's a brief sketch trying to illustrate my situation. Note that each Y observation is a distribution (not just the one I pointed out).

EDIT

If it helps, I could also make X a distribution rather than a single value, but it would not be normally distributed. For each observation Y, there are 100s to 1000s of X observations that I'm averaging to get the single X value to examine in this relationship. Rather than averaging them to get a single X value, I could work with the whole distribution of observations.

If that broadens the range of statistical tests that are available then I can do that too. If it doesn't make a difference, I can just use average values as well.

• Please search our site for "weighted regression". You might also include "variance" as a search term.
– whuber
Dec 21, 2016 at 15:26
• Are you suggesting weighted regression due to my comment on standard deviations varying? Note that I'm referring to the SD for the estimates that compose each individual Y observation. I don't think there's much heteroscedasticity in the regression residuals of Y~X... Dec 21, 2016 at 17:04
• The variances of the Y's will contribute to the variances of the regression residuals. Thus some form of weighted regression looks like a good option.
– whuber
Dec 21, 2016 at 18:17