Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distribution-free or having a specified distribution but with the distribution's parameters unspecified. Nonparametric statistics includes both

descriptive statistics A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and an ...

and statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are violated.

Definitions

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

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 A ranking is a relationship between a set of items such that, for any two items, the first is either "ranked higher than", "ranked lower than" or "ranked equal to" the second. In mathematics, this is known as a weak order or total preorder of ...

but no clear numerical interpretation, such as when assessing preferences. In terms of levels of measurement, non-parametric methods result in

ordinal data Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an ordinal scale, one of four levels of measurement described b ...

. 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 Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...

. 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 Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...

. 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. * A histogram is a simple nonparametric estimate of a probability distribution. *

Kernel density estimation In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on '' kernels'' as ...

is another method to estimate a probability distribution. *

Nonparametric regression Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship ...

and

semiparametric regression In statistics, semiparametric regression includes regression models that combine parametric and nonparametric models. They are often used in situations where the fully nonparametric model may not perform well or when the researcher wants to u ...

methods have been developed based on

kernels Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...

, splines, and

wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...

s. *

Data envelopment analysis Data envelopment analysis (DEA) is a nonparametric method in operations research and economics for the estimation of production frontiers.Charnes et al (1978) DEA has been applied in a large range of fields including international banking, econom ...

provides efficiency coefficients similar to those obtained by multivariate analysis without any distributional assumption. *

KNNs KNNS (1510 AM) is an oldies radio station in Larned, Kansas, near Great Bend. History The Regional Mexican format began in October 2010. Before this, it was an affiliate of ESPN Radio from the spring of 2008 to 2010; before ESPN, it was an old ...

classify the unseen instance based on the K points in the training set which are nearest to it. * A support vector machine (with a Gaussian kernel) is a nonparametric large-margin classifier. * The method of moments with polynomial probability distributions.

Methods

Non-parametric (or distribution-free) inferential statistical methods are mathematical procedures for statistical hypothesis testing which, unlike

parametric statistics Parametric statistics is a branch of statistics which assumes that sample data comes from a population that can be adequately modeled by a probability distribution that has a fixed set of parameters. Conversely a non-parametric model does not as ...

, 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 ) and the sign test by John Arbuthnot (1710) in analyzing the

human sex ratio In anthropology and demography, the human sex ratio is the ratio of males to females in a population. Like most sexual species, the sex ratio in humans is close to 1:1. In humans, the natural ratio at birth between males and females is sligh ...

at birth (see ).

Notes

General references

* Bagdonavicius, V., Kruopis, J., Nikulin, M.S. (2011). "Non-parametric tests for complete data", ISTE & WILEY: London & Hoboken. . * * Gibbons, Jean Dickinson; Chakraborti, Subhabrata (2003). ''Nonparametric Statistical Inference'', 4th Ed. CRC Press. . * also . * 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. * Wasserman, Larry (2007). ''All of Nonparametric Statistics'', Springer. {{isbn, 0-387-25145-6. Statistical inference Robust statistics Mathematical and quantitative methods (economics)