Locally Most Powerful Group-Sequential Tests with Groups of Observations of Random Size: Finite Horizon

Abstract

We consider sequential hypothesis testing based on observations which are received in groups of random size. The observations are supposed independent both within and between the groups, with a distribution depending on a real-valued parameter θ. We suppose that the group sizes are independent and their distributions are known, and that the groups are formed independently from the observations. We are concerned with a problem of testing a simple hypothesis H₀: θ = θ₀ against a composite alternative H₁: θ > θ₀, supposing that no more than a given number of groups will be available (finite horizon). For any (group-)sequential test, we take into account the following three characteristics: its error probability of the first type, the derivative of its power function at θ = θ₀, and the average cost of observations, under some natural assumptions about the cost structure. Under suitable regularity conditions, we characterize the structure of all sequential tests maximizing the derivative of the power function among all (finite-horizon) sequential tests whose error probability of the first type and the average cost of observations do not exceed some prescribed levels.