Abstract <p>Asymptotic behavior of test powers for random-size samples is considered in the problem of testing a simple hypothesis concerning a univariate parameter against a sequence of closely related alternatives. The concept of test power is introduced in this case. Specific tests are asymptotically compared (for normal samples) using the concept of deficiency, which is the additional number of observations required for a competing test to asymptotically achieve the power of the best test. Two examples are considered to illustrate the results obtained. The first example concerns a truncated binomial distribution, and the second example considers a truncated Poisson distribution. These distributions describe random sample sizes.</p>