This page compares the MannWhitneyWilcoxon test against two other nonparametric statistical tests, which are the KolmogorovSmirnov and the Pearson's Chisquare test.
We introduce each test separately in Section 1 to 3, while elaborating more on the chosen test (MannWhitneyWilcoxon, Section 1).
We give a qualitative comparison in Section 4 and demonstrate that the test of our choice is solely sensitive to a difference in two distributions' medians. It does not have an undesired (for our application) sensitivity to differences in shape, which is the case for the other two.
The MWW test was first presented by Wilcoxon in 1945 [1] and two years later discussed by Mann and Whitney [2] on a more solid mathematical basis. The test assess whether one of two random variables is stochastically larger than the other, i.e. whether their medians differ.
Let X1 and X2 be sets of drawings from unknown distributions functions, respectively. The test to assess whether the two underlying random variables are identical is done in three steps:
μ_{T} =  n_{1}*(n_{1} + n_{2} + 1) 
2 
σ_{T}^{2} =  n_{1}*n_{2}*(n_{1} + n_{2} + 1) 
12 
z =  T  μ_{T} 
σ_{T} 
The twosample KS test assesses whether two probability distributions differ or not [3,4]. It is sensitive to location and shape.
Given two drawings X_{1} and X_{2}, the empirical cumulative distributions functions are F_{1}(x) and F_{2}(x), respectively. Then the test statistic is computed as:
D_{n1, n2} = sup_{x}F_{1}(x)  F_{2}(x)
which is the maximum difference between the two cumulative distribution functions along the horizontal xaxis. n_{1}, n_{2} are the cardinalities of X_{1} and X_{2}, respectively. The statistic D_{n1, n2} can be normalized using precomputed tables [4].The Pearson's Chisquare test assesses whether an observed random variable with distribution follows an expected distribution [5].
Let O_{i} and E_{i} be the bins of the observed and expected probability function, respectively. Then the Chisquare test is:
X^{2} =  Σ_{i=1..n}  (O_{i}  E_{i})^{2} 
E_{i} 
If the test statistic is zero, the respective graph is marked with a dashed frame. One sees that the MWW test is only unequal to zero in the first case where the medians are different. The KS test measures the difference in shape for the example in row two. However, it barely measure the difference in shape in the third row since the cumulative distribution functions are very similar. The test statistic is close to zero. The Chisquare test also measures the difference in the third row since it sums up the squared differences in every single bin.
The semantic graylevel enhancement and color transfer are based on tonemapping curves that adaptively de or increase pixel values in different channles independent of their distribution. As we do not consider the shape of the distribution as an extra feature we use the MWW test. We have found the MWW to be more robust and adapted to our application.


























[1] Frank Wilcoxon, Individual Comparisons by Ranking Methods, Biometrics Bulletin, vol. 1, nr. 6, pp. 8083, 1945
[2] H. B. Mann and D. R. Whitney, On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other, The Annals of Mathematical Statistics, vol. 18, nr. 1, pp. 5060, 1947
[3] A. Kolmogorov, Sulla determinazione empirica di una legge di distributione, Giornale dell' Istituto Italiano degli Attuari 4, pp 8391, 1933
[4] N. Smirnov, Table for Estimating the Goodness of Fit of Empirical Distributions, The Annals of Mathematical Statistics, vol. 19, nr. 2, pp. 279281, 1948
[5] R. L. Plackett, Karl Pearson and the ChiSquared Test, International Statistical Review, vol. 51, nr. 1, pp. 5972, 1983