Testing for equal distributions

Consider the null hypothesis that two random variables, X and Y, have the same probability distributions: ${\displaystyle \mu =\nu }$ . For statistical samples from X and Y:

${\displaystyle x_{1},\dots ,x_{n}}$ and ${\displaystyle y_{1},\dots ,y_{m}}$,

the following arithmetic averages of distances are computed between the X and the Y samples:

${\displaystyle A:={\frac {1}{nm}}\sum _{i=1}^{n}\sum _{j=1}^{m}\|x_{i}-y_{j}\|,B:={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\|x_{i}-x_{j}\|,C:={\frac {1}{m^{2}}}\sum _{i=1}^{m}\sum _{j=1}^{m}\|y_{i}-y_{j}\|}$.

The E-statistic of the underlying null hypothesis is defined as follows:

${\displaystyle E_{n,m}(X,Y):=2A-B-C}$

One can prove^[8][9] that ${\displaystyle E_{n,m}(X,Y)\geq 0}$ and that the corresponding population value is zero if and only if X and Y have the same distribution (${\displaystyle \mu =\nu }$). Under this null hypothesis the test statistic

${\displaystyle T={\frac {nm}{n+m}}E_{n,m}(X,Y)}$

converges in distribution to a quadratic form of independent standard normal random variables. Under the alternative hypothesis T tends to infinity. This makes it possible to construct a consistent statistical test, the energy test for equal distributions.^[10]

The E-coefficient of inhomogeneity can also be introduced. This is always between 0 and 1 and is defined as

${\displaystyle H={\frac {D^{2}(F_{X},F_{Y})}{2\operatorname {\operatorname {E} } \|X-Y\|}}={\frac {2\operatorname {E} \|X-Y\|-\operatorname {E} \|X-X'\|-\operatorname {E} \|Y-Y'\|}{2\operatorname {\operatorname {E} } \|X-Y\|}},}$

where ${\displaystyle \operatorname {E} }$ denotes the expected value. H = 0 exactly when X and Y have the same distribution.

Goodness-of-fit

A multivariate goodness-of-fit measure is defined for distributions in arbitrary dimension (not restricted by sample size). The energy goodness-of-fit statistic is

${\displaystyle Q_{n}=n\left({\frac {2}{n}}\sum _{i=1}^{n}\operatorname {E} \|x_{i}-X\|^{\alpha }-\operatorname {E} \|X-X'\|^{\alpha }-{\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\|x_{i}-x_{j}\|^{\alpha }\right),}$

where X and X' are independent and identically distributed according to the hypothesized distribution, and ${\displaystyle \alpha \in (0,2)}$. The only required condition is that X has finite ${\displaystyle \alpha }$ moment under the null hypothesis. Under the null hypothesis ${\displaystyle \operatorname {E} Q_{n}=\operatorname {E} \|X-X'\|^{\alpha }}$, and the asymptotic distribution of Q_n is a quadratic form of centered Gaussian random variables. Under an alternative hypothesis, Q_n tends to infinity stochastically, and thus determines a statistically consistent test. For most applications the exponent 1 (Euclidean distance) can be applied. The important special case of testing multivariate normality^[9] is implemented in the energy package for R. Tests are also developed for heavy tailed distributions such as Pareto (power law), or stable distributions by application of exponents in (0,1).