# MLE like method for uncertain data

I have a rather basic question for which I could not find an answer for. I want to use a similar technique as Maximum Likelihood Estimator (MLE) or even MLE itself for data which has uncertainties for each observation. If we look at a one dimensional case, it basically means that we have n observations $$X_1, X_2, ... , X_n$$ which are all random variables with $$X_i \sim N(\mu_i, \sigma_i)$$ where the means and standard deviations are known. We can assume that the uncertainties and all the measurements are independent from each other. We can think of these uncertainties as measurement errors. Now I would like to find the distribution $$N(\hat{\mu}, \hat{\sigma})$$ that is most likely to generated this data. Is there a way to generalize the MLE to this problem, where each data point is a distribution?

(I think it should be easier with the Bayesian framework with priors and updating, but I would like to find a frequentist solution)

• Have you looked into MLE estimation for measurement error models?
– Eli
Feb 3 at 16:03
• I can only find materials which deals with regression with errors like Errors-in-variables models, but I couldn't find anything which deals with parametric density estimation.
– zerz
Feb 4 at 9:30
• Oh, I misunderstood. This sounds like measurement error deconvolution. ncbi.nlm.nih.gov/pmc/articles/PMC3100171/….
– Eli
Feb 4 at 14:51