Non-parametric statistics is a branch of statistics concerned with non-parametric statistical models and non-parametric inference, including non-parametric statistical tests. Nonparametric methods are often referred to as distribution free methods as they do not rely on assumptions that the data are drawn from a given probability distribution.

The term non-parametric statistic can also refer to a statistic (a function on a sample) whose interpretation does not depend on the population fitting any parametrized distributions. Order statistics are one example of such a statistic that plays a central role in many non-parametric approaches.

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 has a ranking but no clear numerical interpretation, such as when assessing preferences.

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.

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 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.

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

• Anderson-Darling test
• Cochran's Q
• Cohen's kappa
• Efron-Petrosian test
• Friedman two-way analysis of variance by ranks
• Kendall's tau
• Kendall's W
• Kolmogorov-Smirnov test
• Kruskal-Wallis one-way analysis of variance by ranks
• Kuiper's test
• Mann-Whitney U or Wilcoxon rank sum test
• Maximum parsimony for the development of species relationships using computational phylogenetics
• median test
• Pitman's permutation test
• Rank products
• Siegel-Tukey test
• Spearman's rank correlation coefficient
• Student-Newman-Keuls (SNK) test
• Van Elteren stratified Wilcoxon Rank Sum Test
• Wald-Wolfowitz runs test
• Wilcoxon signed-rank test.

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.

General References

• Wasserman, Larry, "All of Nonparametric Statistics", Springer (2007) (ISBN: 0387251456)
• Gibbons, Jean Dickinson and Chakraborti, Subhabrata, "Nonparametric Statistical Inference", 4th Ed. CRC (2003) (ISBN: 0824740521)

See also

• Parametric statistics
• Resampling (statistics)
• Robust statistics
• Particle filter for the general theory of sequential Monte Carlo methods