On asymptotic minimaxity of kernel based tests
- Ermakov, Mikhail S.
2010 Mathematics Subject Classification
- 62G10 62G20
- Nonparametric hypothesis testing, kernel-based-tests, goodness-of-fit, efficiency, kernel estimator
In the problem of signal detection in Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal $L_2$-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that $L_2$-norms of signal smoothed by the kernels exceed some constants $rho_epsilon > 0$. The constant $rho_epsilon$ depends on the power $epsilon$ of noise and $rho_epsilon to 0$ as $epsilon to 0$. Similar statements are proved also if an additional information on a signal smoothness is given. By theorems on asymptotic equivalence of statistical experiments these results are extended on the problems of testing nonparametric hypothesis on density and regression. The exact asymptotically minimax lower bounds of type II error probabilities are pointed out for all these settings. Similar results are also obtained in the problems of testing parametric hypothesis versus nonparametric sets of alternatives.