Determining the probability distribution by a statistics distribution

Yu. V. Linnik

Teoriâ veroâtnostej i ee primeneniâ, Tome 1 (1956) no. 4, pp. 466-478

Résumé

Let $X$ be a real random variable with the distribution function $F(x)=\mathbf P(X$ 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 $F_Q(x)=\mathbf P(Q$, 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 $.