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)