The dissertation considers construction of confidence intervals for a cumulative distribution function F(z) and its inverse at some fixed points z and u on the basis of an i.i.d. sample where the sample size is relatively small. The sample is modeled as having the flexible Generalized Gamma distribution with all three parameters being unknown. This approach can be viewed as an alternative to nonparametric techniques which do not specify distribution of X and lead to less efficient procedures. The confidence intervals are constructed by objective Bayesian methods and use the Jeffreys noninformative prior. Performance of the resulting confidence intervals is studied via Monte Carlo simulations and compared to the performance of nonparametric confidence intervals based on binomial proportion. In addition, techniques for change point detection are analyzed and further evaluated via Monte Carlo simulations. The effect of a change point on the interval estimators is studied both analytically and via Monte Carlo simulations.