Sign test

A non-parametric test for a hypothesis , according to which a random variable has a binomial distribution with parameters . If the hypothesis is true, then

where

and is the beta-function. According to the sign test with significance level , , the hypothesis is rejected if

where , the critical value of the test, is the integer solution of the inequalities

The sign test can be used to test a hypothesis according to which the unknown continuous distribution of independent identically-distributed random variables is symmetric about zero, i.e. for any real ,

In this case the sign test is based on the statistic

which is governed by a binomial law with parameters if the hypothesis is true.

Similarly, the sign test is used to test a hypothesis according to which the median of an unknown continuous distribution to which independent random variables are subject is ; to this end one simply replaces the given random variables by .

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)