# Distributed Detection with Empirically Observed Statistics

## TL;DR

This paper investigates distributed detection when the underlying distributions are unknown, using noisy empirical statistics, deriving optimal error exponents, and showing that a single channel becomes optimal as training length increases.

## Contribution

It introduces a framework for distributed detection with empirically observed statistics and derives the optimal error exponents, extending classical detection results to unknown distributions.

## Key findings

- Optimal type-II error exponent derived for binary detection.
- Using one channel is optimal as training length ratio tends to infinity.
- Numerical evidence suggests one channel remains optimal for finite training lengths.

## Abstract

Consider a distributed detection problem in which the underlying distributions of the observations are unknown; instead of these distributions, noisy versions of empirically observed statistics are available to the fusion center. These empirically observed statistics, together with source (test) sequences, are transmitted through different channels to the fusion center. The fusion center decides which distribution the source sequence is sampled from based on these data. For the binary case, we derive the optimal type-II error exponent given that the type-I error decays exponentially fast. The type-II error exponent is maximized over the proportions of channels for both source and training sequences. We conclude that as the ratio of the lengths of training to test sequences $\alpha$ tends to infinity, using only one channel is optimal. By calculating the derived exponents numerically, we conjecture that the same is true when $\alpha$ is finite. We relate our results to the classical distributed detection problem studied by Tsitsiklis, in which the underlying distributions are known. Finally, our results are extended to the case of $m$-ary distributed detection with a rejection option.