Nonparametric statistics

In statistics, the term non-parametric statistics has at least two different meanings:

1. The first meaning of non-parametric 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 non-parametric statistical models, inference and statistical tests.
   • non-parametric 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 non-parametric approaches.

2. The second meaning of non-parametric 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:

   • non-parametric regression, which refers to modeling where 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 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 but no clear numerical interpretation, such as when assessing preferences; in terms of levels of measurement, for data on an ordinal scale.

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.

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.

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 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

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
• Statistical Bootstrap Methods
• Cochran's Q
• Cohen's kappa
• Friedman two-way analysis of variance by ranks
• Kaplan–Meier
• Kendall's tau
• Kendall's W
• Kolmogorov–Smirnov test
• Kruskal-Wallis one-way 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 signed-rank test.

See also

• Parametric statistics
• Resampling (statistics)

General references

• Corder, G.W. & Foreman, D.I. (2009) Nonparametric Statistics for Non-Statisticians: A Step-by-Step 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. Vol. 5 (First ed.). London: Edward Arnold. pp. xiv+467 pp. ISBN 0-340-54937-8, 0-471-19479-4. MR 1604954. {{cite book}}: Check |isbn= value: invalid character (help); Cite has empty unknown parameter: |1= (help); Unknown parameter |location2= ignored (help); Unknown parameter |publisher2= ignored (help)
• Wasserman, Larry (2007) All of nonparametric statistics, Springer. ISBN 0387251456
• Bagdonavicius, V., Kruopis, J., Nikulin, M.S. (2011). "Non-parametric tests for complete data", ISTE&WILEY: London&Hoboken. ISBN 9781848212695

External links

• Non-parametric estimation Including the saturated model