Minimax nonparametric testing of hypotheses on the distribution density

Let $X_1,\dots,X_n$ be independent identically distributed random variables having unknown density $f(x)$ in $L_2(\nu)$. The problem consists in testing the hypothesis $f(x)=p(x)$ against the alternative that $f(x)$ belongs to an ellipsoid in $L_2(\nu)$ from which a sphere with center at the point $p(x)$ is removed. To solve the problem we construct an asymptotically minimax sequence of tests. As an example the case where the ellipsoid is a sphere in a Sobolev space is considered.