“Characterization of probability distributions”的意思、由来-开放百科全书

In mathematics in general, a characterization theorem says that a particular object – a function, a space, etc. – is the only one that possesses properties specified in the theorem. A characterization of a probability distribution accordingly states that it is the only probability distribution that satisfies specified conditions. More precisely, the model of characterization of

probability distribution was described by [https://ru.wikipedia.org/wiki/%D0%97%D0%BE%D0%BB%D0%BE%D1%82%D0%B0%D1%80%D1%91%D0%B2,_%D0%92%D0%BB%D0%B0%D0%B4%D0%B8%D0%BC%D0%B8%D1%80_%D0%9C%D0%B8%D1%85%D0%B0%D0%B9%D0%BB%D0%BE%D0%B2%D0%B8%D1%87 V.M. Zolotarev] ^[1] in such manner. On the probability space we define the space

of random variables with values in measurable metric space

and the space

of random variables with values in measurable metric space

. By characterizations of probability distributions we understand general problems of description of some set

in the space

by extracting the sets

and

which describe the properties of random variables

and their images

, obtained by means of a specially chosen mapping

The description of the properties of the random variables

and of their images

is equivalent to the indication of the set

from which

must be taken and of the set

into which its image must fall. So, the set which interests us appears therefore in the following form:

where

denotes the complete inverse image of

. This is the general model of characterization of probability distribution. Some examples of characterization theorems:

Verification of conditions of characterization theorems in practice is possible only with some error

, i.e., only to a certain degree of accuracy.^[5] Such a situation is observed, for instance, in the cases where a sample of finite size is considered. That is why there arises the following natural question. Suppose that the conditions of the characterization theorem are fulfilled not exactly but only approximately. May we assert that the conclusion of the theorem is also fulfilled approximately? The theorems in which the problems of this kind are considered are called stability characterizations of probability distributions.

See also

References

1. ^[https://www.researchgate.net/publication/231029496 V.M. Zolotarev (1976). Metric distances in spaces of random variables and their distributions. Matem. Sb., 101 (143), 3 (11)(1976)]
2. ^¹A. M. Kagan, Yu. V. Linnik and C. Radhakrishna Rao (1973). [https://www.researchgate.net/publication/44720749_Characterization_problems_in_mathematical_statistics_by_A_M_Kagan_Yu_V_Linnik_and_C_Radhakrishna_Raotranslated_from_Russian_text_by_B_Ramachandran Characterization Problems in Mathematical Statistics]. John Wiley and Sons, New York, XII+499 pages.
3. ^Pólya, Georg (1923)."Herleitung des Gaußschen Fehlergesetzes ans einer Funktionalgleichung". Mathematische Zeitschrift. 18: 96–108. {{ISSN|0025-5874}}; 1432–1823.
4. ^R. Yanushkevichius.[https://www.academia.edu/24648381/R.Yanushkevichius._Stability_characterizations_of_distributions._1 Stability for characterizations of distributions.] Vilnius, Mokslas, 1991.
5. ^R. Yanushkevichius.[https://www.academia.edu/24648382/R.Yanushkevichius._Stability_characterizations_of_distributions._2 Stability characterizations of some probability distributions.] Saarbrücken, LAP LAMBERT Academic Publishing, 2014.