On the location parameter confidence intervals based on a random size sample from a partially known population

The problem of constructing confidence intervals of a fixed length for the location parameter based on a random size sample is considered. It is proposed to use the confidence interval
$$ \theta^*-u\sqrt p/\sigma<\theta<\theta^*+u\sqrt p/\sigma, $$
where $\theta^*$ is an adaptive estimator, $\sigma^2$ is the Fisher information, and $p^{-1}$ is the mean of the sample size. Nonparametric bounds are given for the limit as $p\to0$ confidence probability. Bibliography: 5 titles.