# Nonparametric statistics

Nonparametric statistics are statistics not based on parameterized families of probability distributions. They include both descriptive and inferential statistics. The typical parameters are the mean, variance, etc. Unlike parametric statistics, nonparametric statistics make no assumptions about the probability distributions of the variables being assessed. The difference between parametric models and non-parametric models is that the former has a fixed number of parameters, while the latter grows the number of parameters with the amount of training data. Note that the non-parametric model does, counterintuitively, contain parameters: the distinction is that parameters are determined by the training data in the case of non-parametric statistics, not the model.

## Definitions

In statistics, the term "non-parametric statistics" has at least two different meanings:

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

The following discussion is taken from Kendall's.

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.

2. 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:
• non-parametric regression, which refers to modeling where 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.

## Applications and purpose

Non-parametric methods are widely used for studying populations that take on a ranked order (such as movie reviews receiving one to four stars). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of measurement, non-parametric methods result in "ordinal" data.

As non-parametric methods make fewer assumptions, their applicability is much wider than the corresponding parametric methods. In particular, they may be applied in situations where less is known about the application in question. Also, due to the reliance on fewer assumptions, non-parametric methods are more robust.

Another justification for the use of non-parametric methods is simplicity. In certain cases, even when the use of parametric methods is justified, non-parametric methods may be easier to use. Due both to this simplicity and to their greater robustness, non-parametric methods are seen by some statisticians as leaving less room for improper use and misunderstanding.

The wider applicability and increased robustness of non-parametric tests comes at a cost: in cases where a parametric test would be appropriate, non-parametric tests have less power. In other words, a larger sample size can be required to draw conclusions with the same degree of confidence.

## Non-parametric models

Non-parametric models differ from parametric models in that the model structure is not specified a priori but is instead determined from data. The term non-parametric is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance.

## Methods

Non-parametric (or distribution-free) inferential statistical methods are mathematical procedures for statistical hypothesis testing which, unlike parametric statistics, make no assumptions about the probability distributions of the variables being assessed. The most frequently used tests include

## Notes

1. Murphy, Kevin (2012). Machine Learning: A Probabilistic Perspective. MIT. p. 16. ISBN 978-0262018029.
2. Stuart A., Ord J.K, Arnold S. (1999), Kendall's Advanced Theory of Statistics: Volume 2A—Classical Inference and the Linear Model, sixth edition, §20.2–20.3 (Arnold).

## General references

• Bagdonavicius, V., Kruopis, J., Nikulin, M.S. (2011). "Non-parametric tests for complete data", ISTE & WILEY: London & Hoboken. ISBN 978-1-84821-269-5.
• Corder, G. W.; Foreman, D. I. (2014). Nonparametric Statistics: A Step-by-Step Approach. Wiley. ISBN 978-1118840313.
• Gibbons, Jean Dickinson; Chakraborti, Subhabrata (2003). Nonparametric Statistical Inference, 4th Ed. CRC Press. ISBN 0-8247-4052-1.
• Hettmansperger, T. P.; McKean, J. W. (1998). Robust nonparametric statistical methods. Kendall's Library of Statistics. 5 (First ed.). London: Edward Arnold. New York: John Wiley & Sons. ISBN 0-340-54937-8. MR 1604954. also ISBN 0-471-19479-4.
• Hollander M., Wolfe D.A., Chicken E. (2014). Nonparametric Statistical Methods, John Wiley & Sons.
• Wasserman, Larry (2007). All of Nonparametric Statistics, Springer. ISBN 0-387-25145-6.