What is the difference between distribution free statistics/methods and non-parametric statistics? 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:

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*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!
 A: 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.
A: The text on Wikipedia has since been revised, and in my opinion makes more sense now. In particular, it quotes Kendall on a possible distinction between nonparametric and distribution free, which however has not been adopted:

The term "nonparametric statistics" has been imprecisely defined in the following two ways, among others.

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*The first meaning of nonparametric covers techniques that do not rely on data belonging to any particular parametric family of probability distributions.
These include, among others:

*

*distribution free methods, which do not rely on assumptions that the data are drawn from a given parametric family of probability distributions. As such it is the opposite of parametric statistics.
nonparametric statistics (a statistic is defined to be a function on a sample; no dependency on a parameter).

Order statistics, which are based on the ranks of observations, is one example of such statistics.
The following discussion is taken from Kendall's.[2]

Statistical hypotheses concern the behavior of observable random variables.... For example, the hypothesis (a) that a normal distribution has a specified mean and variance is statistical; so is the hypothesis (b) that it has a given mean but unspecified variance; so is the hypothesis (c) that a distribution is of normal form with both mean and variance unspecified; finally, so is the hypothesis (d) that two unspecified continuous distributions are identical.
It will have been noticed that in the examples (a) and (b) the distribution underlying the observations was taken to be of a certain form (the normal) and the hypothesis was concerned entirely with the value of one or both of its parameters. Such a hypothesis, for obvious reasons, is called parametric.
Hypothesis (c) was of a different nature, as no parameter values are specified in the statement of the hypothesis; we might reasonably call such a hypothesis non-parametric. Hypothesis (d) is also non-parametric but, in addition, it does not even specify the underlying form of the distribution and may now be reasonably termed distribution-free. Notwithstanding these distinctions, the statistical literature now commonly applies the label "non-parametric" to test procedures that we have just termed "distribution-free", thereby losing a useful classification.



*The second meaning of non-parametric covers techniques that do not assume that the structure of a model is fixed. Typically, the model grows in size to accommodate the complexity of the data. In these techniques, individual variables are typically assumed to belong to parametric distributions, and assumptions about the types of connections among variables are also made. These techniques include, among others:

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*non-parametric regression, which is modeling whereby the structure of the relationship between variables is treated non-parametrically, but where nevertheless there may be parametric assumptions about the distribution of model residuals.

*non-parametric hierarchical Bayesian models, such as models based on the Dirichlet process, which allow the number of latent variables to grow as necessary to fit the data, but where individual variables still follow parametric distributions and even the process controlling the rate of growth of latent variables follows a parametric distribution.




