On a~density estimation within a~class of entire functions

I. A. Ibragimov ; R. Z. Has'minskiǐ

Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 3, pp. 514-524

Résumé

Let $X_1,\dots,X_n$ be i. i. d. random variables with values in $R^k$ and $p(x)$ be their density. Denote by $\Sigma(\mathbf K)$ the class of density functions such that their characteristic functions have symmetric compact support $\mathbf K$. For an arbitrary estimator $T_n(x)$ consider a function $$ \Delta_n^2(T_n,p)=\mathbf E_p\|T_n-p\|_2^2, $$ where $\|\cdot\|_2$ is the $\mathscr L_2$-norm, $\mathbf E_p(\cdot)$ is the expectation with respect to the measure generated by $X_1,\dots,X_n$. We prove the equality $$ \lim_{n\to\infty}[n\inf_{T_n}\sup_{p\in\Sigma(\mathbf K)}\Delta_n^2(T_n,p)]=\frac{\operatorname{mes}\mathbf K}{(2\pi)^k} $$ and some related results.