A non-parametric test for a hypothesis $ H _ $, according to which a random variable $ \mu $ has a binomial distribution with parameters $ ( n ; p = 0 . 5 ) $. If the hypothesis $ H _ $ is true, then
$$ <\mathsf P>\left \< \mu \leq k \left | n , \frac \right . \right \> = \sum _ ^ < k >\left ( \begin n \\ i \end \right ) \left ( \frac \right ) ^ = \ I _ ( n - k , k + 1 ) , $$
$$ I _ ( a , b ) = \frac \int\limits _ < 0 >^ < z >t ^ ( 1 - t ) ^ dt ,\ \ 0 \leq z \leq 1 , $$
and $ B ( a , b ) $ is the beta-function. According to the sign test with significance level $ \alpha $, $ 0 < \alpha \leq 0 . 5 $, the hypothesis $ H _ $ is rejected if
where $ m = m ( \alpha , n ) $, the critical value of the test, is the integer solution of the inequalities
$$ \sum _ ^ < m >\left ( \begin n \\ i \end \right ) \left ( \frac \right ) ^ \leq \frac \alpha ,\ \ \sum _ ^ < > \left ( \begin n \\ i \end \right ) \left ( \frac \right ) ^ > \frac \alpha . $$
The sign test can be used to test a hypothesis $ H _ $ according to which the unknown continuous distribution of independent identically-distributed random variables $ X _ \dots X _ $ is symmetric about zero, i.e. for any real $ x $,
In this case the sign test is based on the statistic
$$ \mu = \sum _ < i=1>^ < n >\delta ( X _ ) ,\ \ \delta ( x) = \left \< \begin 1 & \textrm < if >x > 0 , \\ 0 & \textrm < if >x < 0 , \\ \end\right .$$
which is governed by a binomial law with parameters $ ( n ; p = 0 . 5 ) $ if the hypothesis $ H _ $ is true.
Similarly, the sign test is used to test a hypothesis $ H _ $ according to which the median of an unknown continuous distribution to which independent random variables $ X _ \dots X _ $ are subject is $ \xi _ $; to this end one simply replaces the given random variables by $ Y _ = X _ - \xi _ ,\dots, Y _ = X _ - \xi _ $.
| [1] | 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) |
| [2] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |
| [3] | B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) |
| [4] | N.V. Smirnov, I.V. Dunin-Barkovskii, "Mathematische Statistik in der Technik" , Deutsch. Verlag Wissenschaft. (1969) (Translated from Russian) |
How to Cite This Entry:
Sign test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sign_test&oldid=49780
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article