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Here's my specific question:

Is there a statistical procedure that compares a hypothesized value to the empirical distribution, rather comparing an empirical value to a null distribution?

Put another way: usually with hypothesis tests we get a sample and compare the resulting sample statistic to our hypothesized null distribution. Is there a procedure in which we gather a sample, then compare a hypothesized value to the empirical distribution resulting from the sample?

This question grew out of this earlier question, if it's helpful to read:

Why shift the mean of a bootstrap distribution when conducting a hypothesis test?

Thanks!

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    $\begingroup$ Are you perhaps referring to tests of distribution? If so, good keywords for a search are "Kolmogorov Smirnov," "Anderson Darling," "Shapiro Wilk," "GoF distribution," and (God help us, because there's never any reason to use it but it's popular) "Jarque Bera." If not, could you clarify what you mean by "hypothesized value"? $\endgroup$ – whuber Jan 23 at 23:27
  • $\begingroup$ @whuber: have you seen the followup question that StatsPlease posted? I'd be interested in your take. $\endgroup$ – S. Kolassa - Reinstate Monica Jan 24 at 7:21
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Interesting question. The first thing that comes to mind and it’s close to what you describe, is a Bayesian kind of approach.

You can imagine having a non-parametric data distribution as your likelihood and then your hypothetical value can become your prior (pseudo or not) distribution. Combining those two, you will end up with a posterior distribution which you can then compare to your hypothetical prior that you had in the first place, and test if these two are the same or test if the prior is contained inside the “main body” of the posterior.

Example: let’s say that you believe that the average height is 1.78m. You can create a hypothetical distribution as your prior for which “most if its mass” is concentrated around 1.78m. Afterwards, you can combine it with the data distribution from your sample and test if the resulting distribution is not significantly different than your prior. The way I would choose to test it is via simulation and graphs! E.g: simulate 1000 values from the prior and overlay them on top of a boxplot from simulated values coming from the posterior.

Not sure if I’m describing something that people are doing already but to me sounds interesting.

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If you compare a hypothesized value with a bootstrapped confidence interval for a parameter, you'd be comparing a hypothesized value to a confidence interval based on an empirical distribution from bootstrapping.

This is something that's done -- to see if a hypothesized value is within the confidence interval as form of hypothesis test.

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