In this thesis, we explore the structure of consistent bootstrap statistics in hypothesis testing. Bootstrap, as a very useful technique when theoretical distributions are not available or when the sample size is small, enjoys a lot of interest from applied statisticians. Historically, guidelines for performing Bootstrap have been proposed. One of the guidelines proposed is to center the bootstrap statistic around the true statistic, calculated from the original sample. The second, is to perform resampling in a way such that the new sample reflects the hypothesis tested. However, both of the guidelines are proposed based mostly on an empirical point of view. In this project, we show that the calculation of the bootstrap statistic is directly related to the way the new sample is generated. We describe the specific conditions under which the Bootstrap statistic should or should not be centered around the true. As mentioned the resampling scheme that is picked directly influences this choice. The motivation is derived from the independence test and the same arguments apply to the regression slope test. Finally, we provide a generalized setting where a consistent bootstrap statistic is provided, based on the resampling scheme that is picked.

In this thesis we consider orbital stability of certain patterns in stochastic partial differential equations. We study two examples: a rotating wave in a two-dimensional reaction-diffusion equation and a soliton in a parametrically forced nonlinear Schrödinger equation. In both cases, we show that, for small noise, solutions to the stochastic equations remain close to a version of the pattern which is shifted according to some stochastic phase. We give explicit expressions for this phase, and show that it is optimal to first order in the strength of the noise.

To show stability, we construct a multiscale expansion of the solution around an arbitrarily shifted version of the pattern, and show that this expansion is accurate to second order. From this expansion an obvious candidate for the correct phase shift arises. For technical reasons we then construct a sequence of approximations to this phase shift, which is necessary to show the multiscale expansion around the correctly shifted pattern. We then combine this expansion with a deterministic stability result to get stochastic stability.

Finally, we take first steps towards formulating and proving the same results in a more general setting, where the pattern shift is represented by the action of a Lie group. We obtain some estimates necessary for the multiscale expansion, find the correct phase, and formulate necessary assumptions for the stability to hold.

In this thesis, the repetition code for bit flip errors is examined. Based the stabilizer measurements outcome of a run of the repetition code, one does not know exactly which errors have occurred. Statistics can be used to estimate the probability of all possible error events. This probability estimation is investigated for simulated data assuming phenomenological and circuit level noise. Experiments on the repetition code are performed by the DiCarlo group at the Delft University of Technology on a superconducting quantum computer. For this data, some error probabilities are estimated to be nonphysical. The goal of this thesis is to investigate these nonphysical probabilities and to determine whether they result from sampling noise or a problem with the assumed error model that determines the estimation of the probabilities. By estimating the standard deviation using the bootstrap method it is shown that the nonphysical probabilities are not due to sampling noise. Therefore, it is possible that non-conventional errors, such as leakage or crosstalk, are affecting these estimates.

In a further analysis on the error probabilities and their standard deviation, it is shown that the standard error for space and time edges in the experiment is consistent with the standard error from phenomenological noise initialised with the average error data and ancilla error probabilities, even when
these may be affected by non-conventional errors.