Open Collections

Abstract

Practical problems often drive the development of new statistical methods by presenting real-world challenges. Testing the homogeneity of n independent Poisson means when only one observation per population is available is considered in this paper. This scenario is common in fields where limited data from multiple sources must be analyzed to determine whether different groups share the same underlying event rate or mean. These settings often exhibit underlying structural or spatial symmetries that influence statistical behavior. Traditional methods that rely on large sample sizes are not applicable. Hence, it is crucial to develop techniques tailored to the constraints of single observations. Under the null hypothesis, with large n and a fixed common mean λ, the likelihood ratio test statistic (LRTS) is shown to be asymptotically normally distributed, with the mean and variance being approximated by a truncation method and a parametric bootstrap method. Moreover, with fixed n and large λ, under the null hypothesis, the LRTS is shown to be asymptotically distributed as a chi-square with n−1 degrees of freedom. The Bartlett correction method is applied to improve the accuracy of the asymptotic distribution of the LRTS. We highlight the practical relevance of the proposed method through applications to wildfire and radioactive event data, where correlated observations and sparse sampling are common. Simulation studies further demonstrate the accuracy and robustness of the test under various scenarios, making it well-suited for modern applications in environmental science and risk assessment.