Namespaces
Variants
Actions

Difference between revisions of "Non-parametric test"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
(details)
 
Line 1: Line 1:
 
{{TEX|done}}
 
{{TEX|done}}
A [[Statistical test|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|Kolmogorov test]] is a classic example of a non-parametric test. See also [[Non-parametric methods in statistics|Non-parametric methods in statistics]]; [[Kolmogorov–Smirnov test|Kolmogorov–Smirnov 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 test" . The [[Kolmogorov test]] is a classic example of a non-parametric test.
 +
 
 +
See also [[Non-parametric methods in statistics]]; [[Kolmogorov–Smirnov test]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.R. Rao,  "Linear statistical inference and its applications" , Wiley  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.A. Ibragimov,  R.Z. [R.Z. Khas'minskii] Has'minskii,  "Statistical estimation: asymptotic theory" , Springer  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1979)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  C.R. Rao,  "Linear statistical inference and its applications" , Wiley  (1965)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  I.A. Ibragimov,  R.Z. [R.Z. Khas'minskii] Has'minskii,  "Statistical estimation: asymptotic theory" , Springer  (1981)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1979)</TD></TR>
 +
</table>

Latest revision as of 09:06, 10 April 2023

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 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
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article