In this paper is studied a technique based on samples of the form $N, X_1, X_2, \cdots, X_N$ where $N$ has a Poisson distribution, and each $X_i$ has the same continuous distribution function. Such samples, rather than fixed number samples, are appropriate for fixed time period observations where the number of occurrences is a Poisson variate, and are used in biology, insurance, and telephone engineering. We shall introduce a one-sided Kac statistic which is similar to the one-sided Kolmogorov statistic, derive forms for its finite dimensional and asymptotic distributions, find a lower bound for the power of the test, and prove that the test is "modified" consistent. Tabulations of the distributions will be given.
"A Statistical Test Involving a Random Number of Random Variables." Ann. Math. Statist. 37 (5) 1305 - 1311, October, 1966. https://doi.org/10.1214/aoms/1177699274