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Is it possible to construct a parameter distribution (e.g. Normal or StudentT) from the Estimate and Standard Error of a NonlinearModelFit result? The SE is the square root of the variance but what is the assumption of the underlying parameter distribution and how would the SE feature?

I gather it is possible if one uses Bayesian fitting(e.g. ResourceFunction["BayesianLinearRegression"]) but what about the standard maximum likelihood method that (I assume) NonlinearModelFit uses?

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    $\begingroup$ I think you're mixing Bayesian and Frequentist philosophies in the way the question is written. When using maximum likelihood (a Frequentist approach), there is no distribution for the parameter but there is a distribution for the estimator of a parameter. You might want to ask this question at CrossValidated (stats.stackexchange.com) for clarification and then back here for implementation. $\endgroup$
    – JimB
    Commented Sep 14, 2022 at 16:48
  • $\begingroup$ It is possible using Quantile Regression : mathematicaforprediction.wordpress.com/2014/01/13/… $\endgroup$ Commented Sep 14, 2022 at 17:25
  • $\begingroup$ @AntonAntonov Not true for the distribution of a parameter (unless one goes the reasonable route of a Bayesian approach) but that is true for the distribution of the estimator of a parameter. It might sound like semantics but there's a big difference between a parameter and an estimator of a parameter. $\endgroup$
    – JimB
    Commented Sep 14, 2022 at 17:31
  • $\begingroup$ @JimB Yeah, I misinterpreted OP's NonLinearFit request. That said, these MSE discussions might be more relevant: (1) "Extracting signal from gaussian noise", (2) "Fit function to histogram". $\endgroup$ Commented Sep 14, 2022 at 17:40
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    $\begingroup$ @AntonAntonov For whatever it's worth, quantile regression is way under-used and probably is exactly what folks really need (when the sample size is large enough). $\endgroup$
    – JimB
    Commented Sep 14, 2022 at 17:43

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