# Nonparametric statistics

Nonparametric statistics is a type of statistical analysis that does not rely on the assumption of a specific underlying distribution (such as the normal distribution), or any other specific assumptions about the population parameters (such as mean and variance). This is in contrast to parametric statistics, which make such assumptions about the population. In nonparametric statistics, a distribution may not be specified at all, or a distribution may be specified but its parameters, such as the mean and variance, are not assumed to have a known value or distribution in advance. In some cases, parameters may be generated from the data, such as the median. Nonparametric statistics can be used for descriptive statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated.[1]

## Definitions

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

1. The first meaning of nonparametric involves techniques that do not rely on data belonging to any particular parametric family of probability distributions.

These include, among others:

• Methods which are distribution-free, which do not rely on assumptions that the data are drawn from a given parametric family of probability distributions.
• Statistics defined to be a function on a sample, without dependency on a parameter.

An example is Order statistics, which are based on ordinal ranking of observations.

The discussion following is taken from Kendall's Advanced Theory of Statistics.[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.

2. The second meaning of non-parametric involves 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 associations among variables are also made. These techniques include, among others:
• 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.

## Applications and purpose

Non-parametric methods are widely used for studying populations that have 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 more general 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.

Non-parametric methods are sometimes considered simpler to use and more robust than parametric methods, even when the assumptions of parametric methods are justified. This is due to their more general nature, which may make them less susceptible to misuse and misunderstanding. Non-parametric methods can be considered a conservative choice, as they will work even when their assumptions are not met, whereas parametric methods can produce misleading results when their assumptions are violated.

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 statistical 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

## History

Early nonparametric statistics include the median (13th century or earlier, use in estimation by Edward Wright, 1599; see Median § History) and the sign test by John Arbuthnot (1710) in analyzing the human sex ratio at birth (see Sign test § History).[3][4]

## Notes

1. ^ Pearce, J; Derrick, B (2019). "Preliminary testing: The devil of statistics?". Reinvention: An International Journal of Undergraduate Research. 12 (2). doi:10.31273/reinvention.v12i2.339.
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).
3. ^ Conover, W.J. (1999), "Chapter 3.4: The Sign Test", Practical Nonparametric Statistics (Third ed.), Wiley, pp. 157–176, ISBN 0-471-16068-7
4. ^ Sprent, P. (1989), Applied Nonparametric Statistical Methods (Second ed.), Chapman & Hall, ISBN 0-412-44980-3

## 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. Vol. 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.
• Sheskin, David J. (2003) Handbook of Parametric and Nonparametric Statistical Procedures. CRC Press. ISBN 1-58488-440-1
• Wasserman, Larry (2007). All of Nonparametric Statistics, Springer. ISBN 0-387-25145-6.