From Wikipedia

The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:

  • distribution free methods, which do not rely on assumptions that the data are drawn from a given probability distribution. As such it is the opposite of parametric statistics. It includes non-parametric statistical models, inference and statistical tests.
  • non-parametric statistics (in the sense of a statistic over data, which is defined to be a function on a sample that has no dependency on a parameter), whose interpretation does not depend on the population fitting any parametrized distributions. Statistics based on the ranks of observations are one example of such statistics and these play a central role in many non-parametric approaches.

I can't see the difference between the two cases: distribution free methods, and non-parametric statistics. Do they both not assume the data coming from some distribution? How do they differ?

Thanks and regards!

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    $\begingroup$ The definition you quote suggests the second is a subset of the first, but as they've actually defined them there (I'd swap about some parts of those definitions to the other term!) - and usually in practice - they seem to be used interchangeably. Nonparametric in this sense basically means 'infinite-parametric' while distribution-free methods are ones whose implementation and properties like null distributions don't depend on the distributional shape. Some books do make a distinction between the two; if I think of a reference I'll come back and add it. $\endgroup$ – Glen_b Feb 1 '13 at 0:17
  • $\begingroup$ @Glen_b: Thanks! Some references would be also appreciated! $\endgroup$ – Tim Feb 1 '13 at 13:25
  • $\begingroup$ @Glen_b: Why "the second is a subset of the first"? I feel the opposite. Could you let me know some references? Thanks! $\endgroup$ – Tim Mar 10 '13 at 0:44
  • $\begingroup$ "It includes non-parametric statistical models" is what gives that impression. References on definitions of the terms? Various books on distribution-free/nonparametric stats attempt definitions or distinctions; it's a long time since I read through a bunch of them, but standard books like Conover, Bradley, Daniel, Marascuilo & McSweeney, Lindley would be a start. Of those, I'd be inclined to check Bradley first. I only have Conover and Neave & Worthington to hand; I didn't spot a definition in either in a few minutes of looking - to my surprise; I though both would have something. $\endgroup$ – Glen_b Mar 10 '13 at 1:03
  • $\begingroup$ @Glen_b: Thanks! Do you think any of the two meanings for nonparametric statistics in the quote has something to do with distribution-free statistics? $\endgroup$ – Tim Mar 10 '13 at 1:23

An illustrative example of the difference - comparing samples from two populations.

With the first definition you might still compare the means of the two populations, somehow using the samples to draw inferences (for example, by comparing sample means). The population means are parameters, but you make no assumptions about the distribution (eg you do not assume the population is normally distributed). So this is "distribution free" statistics. Me, I do not think this should be called part of non-parametric statistics - because of the obvious logical contradiction.

Under the second definition you do not consider at all a population mean or any other parameter. Instead you use methods such as comparisons of rankings. This is true non-parametric statistics.

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  • $\begingroup$ Thanks! In both cases, do the distributions of their statistics both not rely on the true distribution of the sample? $\endgroup$ – Tim Mar 10 '13 at 0:34
  • $\begingroup$ Do you agree with Glen_b that "the second is a subset of the first"? $\endgroup$ – Tim Mar 10 '13 at 0:43
  • $\begingroup$ Tim, I don't think the second is a subset of the first; please reread my comment and you'll see that's not at all what I said. I was describing what the thing you quoted appeared to be saying was the case. If I say "It looks like Bill thinks X", it doesn't imply "Glen_b thinks X". I may think nothing of the kind. $\endgroup$ – Glen_b Mar 10 '13 at 1:08
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    $\begingroup$ Irrespective of who (if anyone) thinks so, no, the second case is not a subset of the first. The second case explicitly excludes interest in parameters, which are the focus of the first. $\endgroup$ – Peter Ellis Mar 10 '13 at 1:55
  • $\begingroup$ @PeterEllis That's a good point $\endgroup$ – Glen_b Mar 10 '13 at 2:17

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