# Hellinger distance

A distance between probability measures, expressed in terms of the Hellinger integral. Suppose that on a measurable space a family of probability measures , , is given that are all absolutely continuous relative to some -finite measure on .

The Hellinger distance between two measures and () is defined by the formula

where

is the Hellinger integral. The Hellinger distance does not depend on the choice of the measure and has the following properties:

1) ;

2) if and only if the measures and are mutually singular;

3) if and only if .

Let

be the distance in variation between the measures and . Then

#### References

 [1] H.H. Kuo, "Gaussian measures on Banach spaces" , Springer (1975) [2] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) [3] I.A. Ibragimov, R.Z. [R.Z. Khas'minskii] Has'minskii, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian) [4] V.M. Zolotarev, "Properties and relations of certain types of metrics" Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Akad. Nauk. USSR , 87 (1979) pp. 18–35; 206–212 (In Russian) (English summary)
How to Cite This Entry:
Hellinger distance. M.S. Nikulin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hellinger_distance&oldid=16453
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098