Exact Statistical Inference in Parametric Models - Methods for Constructing Confidence Intervals and Confidence Regions Based on Conditional Parametric Bootstrap and Data Depth
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In this Master's thesis we investigate approaches for constructing approximate and exact confidence intervals and regions in parametric models. A supposedly exact method, called conditional parametric bootstrap, is used to generate confidence intervals for the parameters in the gamma distribution. However, simulation studies are carried out that question the correctness of this method. More precisely, the scale parameter seems to obtain a higher coverage probability than expected. The results are compared to approximate intervals using the more familiar bootstrap methods. Next, we look at a concept called data depth, and apply it on two-dimensional distributions and data sets. This can be used to order multidimensional data, and here we analyze some of the well known types of depths. These different types are then used, in combination with methods from the conditional parametric bootstrap, to construct approximate confidence regions for the parameters in the gamma distribution. The coverage probabilities are analyzed, and we observe how one can obtain close to exact confidence regions just by adjusting a simulation parameter in the algorithm.