  First and second kind modifications of usual confidence intervals for
estimating the expectation and of usual local alternative parameter choices are
introduced in a way such that the asymptotic behavior of the true non-covering
probabilities and the covering probabilities under the modified local non-true
parameter assumption can be asymptotically exactly controlled. The orders of
convergence to zero of both types of probabilities are assumed to be suitably
bounded below according to an Osipov-type condition and the sample distribution
is assumed to satisfy a corresponding tail condition due to Linnik. Analogous
considerations are presented for the power function when testing a hypothesis
concerning the expectation both under the assumption of a true hypothesis as
well as under a modified local alternative. Applications are given for
exponential families.
