Sign test
A non-parametric test for a hypothesis ,
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) |
Sign test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sign_test&oldid=49780