Minimax Nonparametric Testing of Hypotheses on the Distribution Density
Михаил Сергеевич Ермаков · Theory of Probability and Its Applications · 1995
Let $X_1 , \cdots ,X_n $ be independent identically distributed random variables having unknown density $f(x)$ in $L_2 ( u)$. The problem consists in testing the hypothesis $f(x) = p(x)$ against the alternative that $f(x)$ belongs to an ellipsoid in $L_2 ( u)$ 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.