How is a test task being nonparametric or parametric defined?

My understanding is that the test task is parametric, if and only if it assumes a parametric model on the distribution of the sample, regardless of whether its null is true or not. For example, the test task that t tests solve assumes the sample is normally distributed, regardless of whether their nulls are true.

But what if the distribution of the sample isn't assumed to be parametric, but the null specifies the distribution of the sample being parametric? For example,

  • when the null specifies that the distribution of the sample is normal, i.e. testing normality of the data. Is the test task parametric or nonparametric?

  • when the null specifies that the sample has a specific distribution, i.e. goodness-of-fit test, is the testing task parametric or nonparametric? Such as the test tasks that the chi-square test solves, and the test task that the Kolmogorov-Smirnov one-sample test solves?

  • $\begingroup$ (1) What is the distinction between "test" and "test task"? (2) Many authors have characterized the t test as nonparametric because it only assumes the sampling distribution is approximately normal when the null is true. (3) By definition, "hypotheses" are subsets of all possibilities (the "states"). If the set of states is non-parametric, then the test is non-parametric, period. (4) In your second question you appear to confuse the distribution of the data with the distribution of a test statistic. Almost all non-parametric tests have parametric statistical distributions! $\endgroup$ – whuber May 28 '13 at 15:12
  • $\begingroup$ @whuber: Thanks! (1) by "test task", I mean the task a test rule is trying to solve. It includes an assumption i.e. statistical model (parametric or nonparametric) on the distribution of the sample, the null and the alternative. For example, testing normality is a testing task, and goodness-of-fit is another testing task. (4) "chi-square test" is a test rule, solving the test task of goodness-of-fit. Does the test task it tries to solve assume the distribution of the sample to be multinomial, regardless of whether the null is true or not? If yes, isn't the task parametric? $\endgroup$ – Tim May 28 '13 at 15:47
  • $\begingroup$ This is where you confuse the distribution of a test statistic--which usually is parametric (but that fact is irrelevant) with the statistical model. They are not the same! For instance, the Wilcoxon rank-sum test is nonparametric--nobody will legitimately dispute that--but calculation of p-values is made possible because the null distribution of the rank sum statistic on which it is based depends solely on the sizes of the two datasets being compared, and therefore is parametric (for what that's worth). $\endgroup$ – whuber May 28 '13 at 16:24
  • $\begingroup$ @whuber: Yes, I understand the difference between distribution of the test statistic (belonging to the test rule) and the assumption on the distribution of the sample (belonging to the test task). For example, "chi-square test" is a test rule whose test statistic has chi-square distribution under null. It can be used to solve the test task of goodness-of-fit, and does this test task it tries to solve assume the distribution of the sample to be multinomial in the statistical model? If yes, isn't the task parametric? $\endgroup$ – Tim May 28 '13 at 17:09
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    $\begingroup$ If you have questions about what statistical models are, Tim, you can find the answers elsewhere on our site. Comment threads aren't the place for extended chats, so I'm trying to keep this as brief and to the point as possible. I have now repeated at least three times the points I felt were needed to clarify and give context to your question, so I don't see any need to go on. $\endgroup$ – whuber May 28 '13 at 20:49

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