On a sufficient statistics for families of distributions with variable support of density. I

M. S. Ermakov

Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 62-82

Résumé

Let $X_i$, $i=\overline{1,n}$ be independent random vectors with density $f(x,\Theta)$, ($\Theta\in R^d$). Support $f(x,\Theta)$ depend on $\Theta$. Let $R_n\{\Theta|\prod_{i=1}^nf(x_i,\Theta)\ne0\}$. Then $(T,R_n)$ is sufficient statistics, if $\prod_{i=1}^nf(x_i,\Theta)=g_\Theta(T)h(x_1,\dots,x_n)\cdot\chi_{R_n}(\Theta)$, for some measurable functions $g_\Theta(T)$ and $h$. Instead of $(T,R_n)$ we take sufficient statistics $x_{j_1},\dots,x_{j_{\alpha_n}},T$ if $R_n\{\Theta|\prod_{i=1}^{\alpha_n}f(x_{ji},\Theta)\ne0\}$. Denote $x_{j_1},\dots,x_{j_{\alpha_n}}$ suffiсient statistics of described type with minimum $\alpha_n$. Under wide assumptions it is shown, that $\alpha_n$ is bounded on probability when $n\to\infty$ (Th. 4.1). The limit distribution of $\alpha_n$ and $x_{j_1},\dots,x_{j_{\alpha_n}}$ is investigated (Th. 4.4, 4.5). Some weak analogous of Dynkin's theorems is proved for statistics $T$.