Nonparametric statistics

Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as is parametric statistics. 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 five "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's assumptions are met, 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 • Analysis of similarities

• Anderson–Darling test: tests whether a sample is drawn from a given distribution

• Statistical bootstrap methods: estimates the accuracy/sampling distribution of a statistic

• Cochran's Q: tests whether k treatments in randomized block designs with 0/1 outcomes have identical effects

• Cohen's kappa: measures inter-rater agreement for categorical items

• Friedman two-way analysis of variance by ranks: tests whether k treatments in randomized block designs have identical effects

• Empirical likelihood

• Kaplan–Meier: estimates the survival function from lifetime data, modeling censoring

• Kendall's tau: measures statistical dependence between two variables

• Kendall's W: a measure between 0 and 1 of inter-rater agreement.

• Kolmogorov–Smirnov test: tests whether a sample is drawn from a given distribution, or whether two samples are drawn from the same distribution.

• Kruskal–Wallis one-way analysis of variance by ranks: tests whether > 2 independent samples are drawn from the same distribution.

• Kuiper's test: tests whether a sample is drawn from a given distribution, sensitive to cyclic variations such as day of the week.

• Logrank test: compares survival distributions of two right-skewed, censored samples.

• Mann–Whitney U or Wilcoxon rank sum test: tests whether two samples are drawn from the same distribution, as compared to a given alternative hypothesis.

• McNemar's test: tests whether, in 2 × 2 contingency tables with a dichotomous trait and matched pairs of subjects, row and column marginal frequencies are equal.

• Median test: tests whether two samples are drawn from distributions with equal medians.

• Pitman's permutation test: a statistical significance test that yields exact p values by examining all possible rearrangements of labels.

• Rank products: detects differentially expressed genes in replicated microarray experiments.

• Siegel–Tukey test: tests for differences in scale between two groups.

• Sign test: tests whether matched pair samples are drawn from distributions with equal medians.

• Spearman's rank correlation coefficient: measures statistical dependence between two variables using a monotonic function.

• Squared ranks test: tests equality of variances in two or more samples.

• Tukey–Duckworth test: tests equality of two distributions by using ranks.

• Wald–Wolfowitz runs test: tests whether the elements of a sequence are mutually independent/random.

• Wilcoxon signed-rank test: tests whether matched pair samples are drawn from populations with different mean ranks.

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 ).

General references

 * Bagdonavicius, V., Kruopis, J., Nikulin, M.S. (2011). "Non-parametric tests for complete data", ISTE & WILEY: London & Hoboken. ISBN 978-1-84821-269-5.
 * Gibbons, Jean Dickinson; Chakraborti, Subhabrata (2003). Nonparametric Statistical Inference, 4th Ed. CRC Press. ISBN 0-8247-4052-1.
 * also ISBN 0-471-19479-4.
 * 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. ISBN 1-58488-440-1
 * Wasserman, Larry (2007). All of Nonparametric Statistics, Springer. ISBN 0-387-25145-6.
 * Wasserman, Larry (2007). All of Nonparametric Statistics, Springer. ISBN 0-387-25145-6.