Necessary and sufficient conditions of uniform consistency are explored. A hypothesis is simple. Nonparametric sets of alternatives are bounded convex sets in $\mathbb{L}_p$, $p >1$ with "small" balls deleted. The "small" balls have the center at the point of hypothesis and radii of balls tend to zero as sample size increases. For problem of hypothesis testing on a density, we show that, for the sets of alternatives, there are uniformly consistent tests for some sequence of radii of the balls, if and only if, convex set is relatively compact. The results are established for problem of hypothesis testing on a density, for signal detection in Gaussian white noise, for linear ill-posed problems with random Gaussian noise and so on.

翻译：本文探讨了一致一致性的充分必要条件。假设为简单假设。非参数备择集是$\mathbb{L}_p$（$p >1$）中删去“小”球后的有界凸集。“小”球的中心位于假设点，且球的半径随样本量增加趋于零。对于密度假设检验问题，我们证明：对于某些球半径序列，当且仅当凸集是相对紧集时，存在关于这些备择集的一致一致检验。该结果建立于密度假设检验、高斯白噪声中的信号检测、含随机高斯噪声的线性不适定问题等场景中。