## 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:
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:
2. Evaluate criterion statistic:
   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:
   where:
   are modified bessel functions.
4. And at last calculate P-value as: