Sign test

A non-parametric test for a hypothesis $ H _ {0} $, according to which a random variable $ \mu $ has a binomial distribution with parameters $ ( n ; p = 0 . 5 ) $. If the hypothesis $ H _ {0} $ is true, then

$$ {\mathsf P} \left \{ \mu \leq k \left | n , \frac{1}{2} \right . \right \} = \sum _ {i = 0 } ^ { k } \left ( \begin{array}{c} n \\ i \end{array} \right ) \left ( \frac{1}{2} \right ) ^ {n} = \ I _ {0,5} ( n - k , k + 1 ) , $$

$$ k = 0 \dots n , $$

where

$$ I _ {z} ( a , b ) = \frac{1}{B ( a , b ) } \int\limits _ { 0 } ^ { z } t ^ {a-1} ( 1 - t ) ^ {b-1} dt ,\ \ 0 \leq z \leq 1 , $$

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 _ {0} $ is rejected if

$$ \min \{ \mu , n - \mu \} \leq m , $$

where $ m = m ( \alpha , n ) $, the critical value of the test, is the integer solution of the inequalities

$$ \sum _ {i = 0 } ^ { m } \left ( \begin{array}{c} n \\ i \end{array} \right ) \left ( \frac{1}{2} \right ) ^ {n} \leq \frac \alpha {2} ,\ \ \sum _ {i = 0 } ^ { {m + 1} } \left ( \begin{array}{c} n \\ i \end{array} \right ) \left ( \frac{1}{2} \right ) ^ {n} > \frac \alpha {2} . $$

The sign test can be used to test a hypothesis $ H _ {0} $ according to which the unknown continuous distribution of independent identically-distributed random variables $ X _ {1} \dots X _ {n} $ is symmetric about zero, i.e. for any real $ x $,

$$ {\mathsf P} \{ X _ {i} < - x \} = {\mathsf P} \{ X _ {i} > x \} . $$

In this case the sign test is based on the statistic

$$ \mu = \sum _ { i=1} ^ { n } \delta ( X _ {i} ) ,\ \ \delta ( x) = \left \{ \begin{array}{ll} 1 & \textrm{ if } x > 0 , \\ 0 & \textrm{ if } x < 0 , \\ \end{array} \right .$$

which is governed by a binomial law with parameters $ ( n ; p = 0 . 5 ) $ if the hypothesis $ H _ {0} $ is true.

Similarly, the sign test is used to test a hypothesis $ H _ {0} $ according to which the median of an unknown continuous distribution to which independent random variables $ X _ {1} \dots X _ {n} $ are subject is $ \xi _ {0} $; to this end one simply replaces the given random variables by $ Y _ {1} = X _ {1} - \xi _ {0} ,\dots, Y _ {n} = X _ {n} - \xi _ {0} $.

References

[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)