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On uniformly consistent tests

Mikhail Ermakov

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.

On uniformly consistent tests

Abstract

Necessary and sufficient conditions of uniform consistency are explored. A hypothesis is simple. Nonparametric sets of alternatives are bounded convex sets in , 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.
Paper Structure (8 sections, 7 theorems, 22 equations)

This paper contains 8 sections, 7 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Uniformly consistent sets
  3. Hypothesis testing on a density
  4. Hypothesis testing on intensity function of Poisson random process
  5. Signal detection in Gaussian white noise
  6. Hypothesis testing on a solution of linear ill-posed problem
  7. Hypothesis testing on an operator known with a random error
  8. Proof of Theorem \ref{['th1']}

Key Result

Theorem 2.1

There is a sequence $\rho_n \to 0$ as $n \to \infty$ such that sets of alternatives $V(\rho_n)$ is uniformly consistent, if and only if, set $U$ are relatively compact in $\mathbb{L}_p$.

Theorems & Definitions (7)

  • Theorem 2.1
  • Corollary 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6