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.
Definitions
The term "nonparametric statistics" has been defined imprecisely in the following two ways, among others:
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.
A histogram is a simple nonparametric estimate of a probability distribution.
Kernel density estimation is another method to estimate a probability distribution.
Nonparametric regression and semiparametric regression methods have been developed based on kernels, splines, and wavelets.
Data envelopment analysis provides efficiency coefficients similar to those obtained by multivariate analysis without any distributional assumption.
KNNs 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, 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 at birth (see ).
See also
CDF-based nonparametric confidence interval
Parametric statistics
Resampling (statistics)
Semiparametric model
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. .
Statistical inference
Robust statistics
Mathematical and quantitative methods (economics)