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I have a sort of strange problem, where my sample is difficult to obtain, but the population distribution is easy to obtain. Specifically, I have obtained a single observation. I would like to know if it comes from a particular population. A closed form for the population distribution is not known, but I can resample it ad infinitum and obtain an empirical distribution to any desirable degree of accuracy. Judging by the pretty-picture method, the distribution is not normal enough to assume normality, but is normal enough for me to wonder if there is something better to use here than Chebyshev's Inequality.

Are there any Nonparametric methods that can be used to test a single observation against an empirical distribution?

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    $\begingroup$ If your null hypothesis is that the sample observation did come from that population then then there is a $5\%$ probability that the sample will be in top tail or bottom tail of the known samples from the population. So take your resamples (one less than a multiple of $40$ might work) and look at the ranked position of the observation you are interested in among these $\endgroup$
    – Henry
    Commented Feb 3, 2020 at 17:38
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    $\begingroup$ Do you have any specific (perhaps intractable) alternative distribution(s) in mind? $\endgroup$
    – jbowman
    Commented Feb 3, 2020 at 17:41
  • $\begingroup$ @jbowman, not really... I could probably fit e.g. a Gamma to it with some success, since it "looks like" a skewed normal and is guaranteed to be > 0. OTOH, I have no a priori reason to expect the population to be gamma, so a nonparametric form would really be preferable. $\endgroup$
    – Him
    Commented Feb 3, 2020 at 17:48
  • $\begingroup$ @jbowman aaaaahhhhhh, I think I take your meaning now. You mean that if I had some other hypothesis for the population distribution besides the one I can empirically determine, I could employ the Likelihood Ratio Test? $\endgroup$
    – Him
    Commented Feb 5, 2020 at 15:51
  • $\begingroup$ Something along those lines, yes. But if you don't, you don't. $\endgroup$
    – jbowman
    Commented Feb 5, 2020 at 16:18

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There is no statistical test because you have a sample, not a sampling distribution. You can compute distance metrics (eg, Z-score in case of normal distributions) but this would not be an inferential test

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  • $\begingroup$ "but this would not be an inferential test" Are you making a semantical claim or an existential one? $\endgroup$
    – Him
    Commented Feb 4, 2020 at 1:33
  • $\begingroup$ An accurate claim. Inferential tests make assumptions that do not accommodate single point estimates. $\endgroup$
    – HEITZ
    Commented Feb 5, 2020 at 5:27
  • $\begingroup$ You mistake my meaning. Do you intend to say that nothing can be logically inferred about the likelihood that a point estimate belongs to a particular population with a known distribution, or do you intend merely to say that such an inference is not properly "statistical inference"? $\endgroup$
    – Him
    Commented Feb 5, 2020 at 14:32
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    $\begingroup$ If it is the former, then I would posit a simpler analogy, wherein I get a single point estimate of "7", and ask the question "what is the likelihood that this is the result of a coin flip?" Clearly some kind of inference can be made here, although perhaps "statistical inference" is not the correct label. $\endgroup$
    – Him
    Commented Feb 5, 2020 at 14:34
  • $\begingroup$ If I have a null hypothesis that my data is distributed Gaussian $(0,1)$, and I observe a single value of $6$, it seems to me I can certainly apply a test and reject the null; the non-rejection region for $H_0$ might be, e.g., $(-1.96, +1.96)$ or some such, but it's still a test. $\endgroup$
    – jbowman
    Commented Feb 5, 2020 at 16:22

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