# Hypothesis Test for Bounds on the Size of Random Defective Set

## TL;DR

This paper introduces a hypothesis testing method for estimating the number of defectives in group testing, using a new decoding algorithm that is shown to be order-optimal and has low complexity.

## Contribution

A novel decoding algorithm for testing hypotheses about the number of defectives, with proven order-optimality and practical advantages over existing methods.

## Key findings

- The proposed algorithm is order-optimal in certain regimes.
- Simulation results show low complexity and small error probability.
- The method effectively estimates the defective set size without prior knowledge.

## Abstract

The conventional model of disjunctive group testing assumes that there are several defective elements (or defectives) among a large population, and a group test yields the positive response if and only if the testing group contains at least one defective element. The basic problem is to find all defectives using a minimal possible number of group tests. However, when the number of defectives is unknown there arises an additional problem, namely: how to estimate the random number of defective elements. n this paper, we concentrate on testing the hypothesis $H_0$: the number of defectives $\le s_1$ against the alternative hypothesis $H_1$: the number of defectives $\ge s_2$. We introduce a new decoding algorithm based on the comparison of the number of tests having positive responses with an appropriate fixed threshold. For some asymptotic regimes on $s_1$ and $s_2$, the proposed algorithm is shown to be order-optimal. Additionally, our simulation results verify the advantages of the proposed algorithm such as low complexity and a small error probability compared with known algorithms.