 Nonparametric statistics

In statistics, the term nonparametric statistics has at least two different meanings:
 The first meaning of nonparametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:

 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 nonparametric statistical models, inference and statistical tests.
 nonparametric 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 nonparametric approaches.

 The second meaning of nonparametric 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:

 nonparametric regression, which refers to modeling where the structure of the relationship between variables is treated nonparametrically, but where nevertheless there may be parametric assumptions about the distribution of model residuals.
 nonparametric 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.

Contents
Applications and purpose
Nonparametric 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 nonparametric 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, nonparametric methods result in "ordinal" data.
As nonparametric 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, nonparametric methods are more robust.
Another justification for the use of nonparametric methods is simplicity. In certain cases, even when the use of parametric methods is justified, nonparametric methods may be easier to use. Due both to this simplicity and to their greater robustness, nonparametric methods are seen by some statisticians as leaving less room for improper use and misunderstanding.
The wider applicability and increased robustness of nonparametric tests comes at a cost: in cases where a parametric test would be appropriate, nonparametric tests have less power. In other words, a larger sample size can be required to draw conclusions with the same degree of confidence.
Nonparametric models
Nonparametric models differ from parametric models in that the model structure is not specified a priori but is instead determined from data. The term nonparametric 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 provides better estimates of the density than histograms.
 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.
Methods
Nonparametric (or distributionfree) 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
 Anderson–Darling test
 Statistical Bootstrap Methods
 Cochran's Q
 Cohen's kappa
 Friedman twoway analysis of variance by ranks
 Kaplan–Meier
 Kendall's tau
 Kendall's W
 Kolmogorov–Smirnov test
 KruskalWallis oneway analysis of variance by ranks
 Kuiper's test
 Logrank Test
 Mann–Whitney U or Wilcoxon rank sum test
 median test
 Pitman's permutation test
 Rank products
 Siegel–Tukey test
 Spearman's rank correlation coefficient
 Wald–Wolfowitz runs test
 Wilcoxon signedrank test.
See also
General references
 Corder, G.W. & Foreman, D.I. (2009) Nonparametric Statistics for NonStatisticians: A StepbyStep Approach, Wiley ISBN 9780470454619
 Gibbons, Jean Dickinson and Chakraborti, Subhabrata (2003) Nonparametric Statistical Inference, 4th Ed. CRC ISBN 0824740521
 Hettmansperger, T. P.; McKean, J. W. (1998). Robust nonparametric statistical methods. Kendall's Library of Statistics. 5 (First ed.). London: Edward Arnold. pp. xiv+467 pp.. ISBN 0340549378, 0471194794. MR1604954.
 Wasserman, Larry (2007) All of nonparametric statistics, Springer. ISBN 0387251456
 Bagdonavicius, V., Kruopis, J., Nikulin, M.S. (2011). "Nonparametric tests for complete data", ISTE&WILEY: London&Hoboken. ISBN 9781848212695
 The first meaning of nonparametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:
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