The behaviour of the second kind error-probability of a uniformly most powerful criterion, when there is a fixed and a small chance of the first type error is analysed in the paper. The asymptotic formula for probability has been obtained using a common model of the likelihood function under formed natural conditions, i.e. local density. At the same time it is shown as a key result and what overall conditions seem to be in specific models (independent observations, non-homogeneous Poisson process, process of renewal).