# On uniformly consistent tests

## TL;DR

This paper investigates the conditions under which uniformly consistent hypothesis tests exist across various statistical models, emphasizing the role of convexity and compactness in the sets of alternatives.

## Contribution

It provides necessary and sufficient conditions for uniform consistency of tests in nonparametric and Gaussian models, linking convexity and compactness to test feasibility.

## Key findings

- Uniformly consistent tests exist if and only if the convex set of alternatives is relatively compact.
- Results apply to density hypothesis testing, signal detection in Gaussian noise, and linear ill-posed problems.
- Small balls around hypotheses shrink as sample size increases, affecting test consistency.

## Abstract

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.