# Non-parametric statistics

## Overview

**Non-Parametric statistics** are statistics where it is not assumed that the population fits any parametrized distributions. Non-Parametric statistics are typically applied to populations that take on a *ranked* order (such as movie reviews receiving one to four stars).

The branch of statistics known as **non-parametric statistics** is concerned with non-parametric statistical models and non-parametric statistical tests.

**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. Nonparametric models are therefore also called *distribution free* or *parameter-free*.

- 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.

**Non-parametric** (or **distribution-free**) **inferential statistical methods** are mathematical procedures for statistical hypothesis testing which, unlike parametric statistics, make no assumptions about the frequency distributions of the variables being assessed. The most frequently used tests include

- binomial test
- Anderson-Darling test
- chi-square test
- Cochran's Q
- Cohen's kappa
- Efron-Petrosian Test
- Fisher's exact 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
- McNemar's test (a special case of the chi-squared test)
- median test
- Pitman's permutation test
- Siegel-Tukey test
- Spearman's rank correlation coefficient
- Student-Newman-Keuls (SNK) test
- Wald-Wolfowitz runs test
- Wilcoxon signed-rank test.

Nonparametric tests have less power than the appropriate parametric tests, but are more robust when the assumptions underlying the parametric test are not satisfied.

## See also

- parametric statistics
- resampling (statistics)
- robust statistics
- particle filter for the general theory of
*Sequential Monte Carlo*methods