A systematic method of the construction of a confidence interval for the difference between two means is given in the exponential and gamma cases. The application of a similar method to the Behrens-Fisher type problem is given, and a numerical comparison of the confidence interval proposed by Akahira (Stat Pap 43:273–284, 2002) with that based on the Fisher-Welch-Wald similar test is also done. Next, in the case when a parameter is assumed to be nonnegative or positive-valued, we consider an interval estimation problem on an unknown parameter based on the observations including errors, and construct combined Bayesian-frequentist confidence intervals for the parameter. Further, in the sequential interval estimation with fixed width of a location parameter based on a sequence of i.i.d. random variables according to a uniform distribution with unknown location and scale parameters, the asymptotic properties of sequential estimation procedures are obtained.