Wilcoxon Signed Rank Test

Wilcoxon Signed Rank Test

Let Y₁, Y₂, ...., Y_n be a set of independent and identically distributed random variables having a probability distribution that is continuous and symmetric, with density function f_Y(y). Formulate the null and alternative hypotheses

Define the sign indecator Z_i, i = 1, 2, . . ., n, as

The Wilcoxon test statistic is calculated as:

If the null hypothesis is true, the mean and variance of the Wilcoxon test statistic are given by :

For n > 12, the z score of W has an approximately standard normal distribution, i.e.,

Therefore, if n is large, H_o will be rejected in favor of H₁ if

Z_W < z_a/2 or Z_W > z_1-a/2