## Investigating the effect of restricting residuals in bootstrap based hypothesis testing

##### Abstract

In this dissertation, model-based bootstrap methods for testing the hypothesis for the population mean, population variance and regression model coe cient parameters are discussed by considering eight di erent ways to approach these tests. The approaches involve how each of the three stages of the model-based bootstrap hypothesis testing procedure are conducted: 1. In the rst stage, residuals are calculated. Here, the residuals can be obtained from either the unrestricted model (with the null hypothesis not enforced), or the restricted model (with the null hypothesis enforced). 2. In the second stage, the bootstrap data are generated. These values can be generated from either the unrestricted model or the restricted model. 3. In the nal stage, the pivotal test statistic is calculated using either a sample estimate of the parameter of interest or the actual hypothesised value of the parameter. This results in eight di erent combinations of approaches. The aims of this dissertation are to determine which approaches are `correct' and which are not, as well as the gains and losses associated with using certain `correct' and `incorrect' approaches. Additionally, it is also of interest to nd out which approaches produce the same results. This problem is rst tackled by deriving theoretical expressions for the bootstrap test statistics for all eight approaches so that it might be clear how these approaches behave in general. Then a Monte Carlo study is conducted to approximate the size and power of the tests for each of the eight di erent approaches. In this study it is found that the only two approaches that work well for all three test scenarios considered are: The approach where one resamples from unrestricted residuals, generates bootstrap data from the restricted model, and uses the hypothesised value in the test statistic, and the approach where one resamples from unrestricted residuals, generates bootstrap data from the unrestricted model and uses the estimated parameter value in the test statistic. The bootstrap test statistics for these two approaches are equivalent for testing both the mean and variance in the location-scale model and for testing the coe cient parameter in the simple linear regression. The simulation studies corroborate these results, where it is found that these approaches always produce adequate sizes and powers for all simulation settings considered. For each of the two models a few other approaches were found to perform reasonably well, but those approaches di ered for the various models, and did not not produce noticeably higher powers or improvements on the size of the tests. The two tests mentioned are therefore the only two recommended approaches for conducting residual-based bootstrap hypothesis testing.