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Teoriya Veroyatnostei i ee Primeneniya, 1956, Volume 1, Issue 4, Pages 466–478 (Mi tvp5013)  

This article is cited in 3 scientific papers (total in 3 papers)

Determining the probability distribution by a statistics distribution

Yu. V. Linnik

Leningrad
Abstract: Let $X$ be a real random variable with the distribution function $F(x)=\mathbf P(X<x)$ and $\vec{\xi}=(x_1,\dots,x_n)$ the corresponding sample of size $n$ ($x_i$ being independent replicas of $X$).
A statistic $Q(\vec{\xi})$ is called definite if it is homogeneous of positive dimension and the level surfaces $Q(\vec{\xi})=\operatorname{const}$ are continuous, piecewise-smooth and star-finite regions. A statistic $Q(\vec{\xi})$ is called defining in a class $K$ of distribution functions $F(x)$, if the distribution $F_Q(x)=\mathbf P(Q<x)$, induced by $F(x)$, determines $F(x)$ in the class $K$. A definite statistic cannot be defining in general for the class $K$ of all distribution functions, but it is defining in certain rather wide classes of symmetric distribution densities. Three theorems are proved to this effect. The problem can be given as a generalization of the classical moment problem, putting $Q(\vec{\xi})=x_1^2+\cdots+x_n^2 $.
Received: 27.10.1956
English version:
Theory of Probability and its Applications, 1956, Volume 1, Issue 4, Pages 422–434
DOI: https://doi.org/10.1137/1101032
Document Type: Article
Language: Russian
Citation: Yu. V. Linnik, “Determining the probability distribution by a statistics distribution”, Teor. Veroyatnost. i Primenen., 1:4 (1956), 466–478; Theory Probab. Appl., 1:4 (1956), 422–434
Citation in format AMSBIB
\Bibitem{Lin56}
\by Yu.~V.~Linnik
\paper Determining the probability distribution by a statistics distribution
\jour Teor. Veroyatnost. i Primenen.
\yr 1956
\vol 1
\issue 4
\pages 466--478
\mathnet{http://mi.mathnet.ru/tvp5013}
\transl
\jour Theory Probab. Appl.
\yr 1956
\vol 1
\issue 4
\pages 422--434
\crossref{https://doi.org/10.1137/1101032}
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  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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