Lehmann-Rosenblatt analysis.

Lehmann-Rosenblatt test is ω2-type distribution free criterion for homogeneity testing.
Let assume that we have N = m + n observations in two samles:
Samples
Like in Kolmogorov test we suppose:
  1. All N observations X and Y are mutually independent.
  2. All X derived from the entire assembly Π1.
  3. All Y derived from the entire assembly Π2.

Method

Will test hypothesis that both populations Π1 and Π2 are identical, i.e. both samples
were derived from the single population. It can be rewrited:
H0: F(x) = G(x) ∀x.
To test our hypothesis we:
  1. Arrange our observations X and Y:
    arrangement
  2. Evaluate criterion statistic:
    Lehmann-rosenblatt stat
    where ri—index number (rang) of yi, sj—index number (rang) of xj in the joint
    static series.
  3. When m and n tend to ∞, distribution of statistic T upon condition, that H0
    hypothesis is true, tends to a1(t) distribution function:
    a1(t)
    where:
    besel
    are modified bessel functions.
  4. And at last calculate P-value as:
    pval