# Non-parametric test

A statistical test of a hypothesis $H_0$: $\theta\in\Theta_0\subset\Theta$ against the alternative $H_1$: $\theta\in\Theta_1=\Theta\setminus\Theta_0$ when at least one of the two parameter sets $\Theta_0$ and $\Theta_1$ is not topologically equivalent to a subset of a Euclidean space. Apart from this definition there is also another, wider one, according to which a statistical test is called non-parametric if the statistical inferences obtained using it do not depend on the particular null-hypothesis probability distribution of the observable random variables on the basis of which one wants to test $H_0$ against $H_1$. In this case, instead of the term "non-parametric test" one speaks frequently of a "distribution-free statistical testdistribution-free test" . The Kolmogorov test is a classic example of a non-parametric test. See also Non-parametric methods in statistics; Kolmogorov–Smirnov test.

#### References

[1] | C.R. Rao, "Linear statistical inference and its applications" , Wiley (1965) |

[2] | L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) |

[3] | I.A. Ibragimov, R.Z. [R.Z. Khas'minskii] Has'minskii, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian) |

[4] | M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979) |

**How to Cite This Entry:**

Non-parametric test.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Non-parametric_test&oldid=34284