On the second order asymptotically minimax estimates

B. Ya. Levit

Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 3, pp. 561-576

Résumé

Let $X_1,\dots,X_n$ be a sequence of independent random variables having Gaussian distribution $\mathscr N(m,\sigma^2)$ with $\sigma^2$ known and unknown mean $m$ subjected to the restriction $|m|<a$. For an arbitrary estimate $T$ ($X_1,\dots,X_n$) and nonnegative even nondecreasing on $R^+$ loss function $l(x)$ satisfying the condition $$ \int e^{-x^2/2}x^2l(x)\,dx<\infty $$ we consider the corresponding risk $$ R(T,l,m)=\mathbf E_ml\biggl(\frac{\sqrt{n}}{\sigma}(T-m)\biggr). $$ It is shown that for $\varepsilon=\sigma/a\sqrt{n}\to 0$ the following asymptotic expansion for the minimax risk holds: $$ \inf_T\sup_{|m|<a}R(T,l,m)=R_0-1/2R_1\pi^2\varepsilon^2+o(\varepsilon^2), $$ where $$ R_0=\frac{1}{\sqrt{2\pi}}\int e^{-x^2/2}l(x)\,dx,\qquad R_1=\frac{1}{\sqrt{2\pi}}\int e^{-x^2/2}(x^2-1)l(x)\,dx. $$ Different estimates are exhibited which are second order asymptotically minimax simultaneously for a large class of loss functions.