In wireless channels and many other channels, interference is a central phenomenon. Mitigating interference is a key to improving system performance. To find the limit of the achievable rates for these channels in the presence of interference, the two-user Gaussian interference channel has been the subject of intensive study in network information theory. However, most current results have been obtained by assuming Gaussian input distributions. While optimal in single-user Gaussian channels, the issue with this assumption is that the Gaussian noise becomes the worst noise when the input distribution is also Gaussian. In this thesis, we propose a class of discrete constellations. We show that this class of constellations can automatically achieve the same sum rates as schemes that treat interference as noise or perform time sharing.

In wireless channels and many other channels, interference is a central phenomenon. Mitigating interference is a key to improving system performance. To find the limit of the achievable rates for these channels in the presence of interference, the two-user Gaussian interference channel has been the subject of intensive study in network information theory. However, most current results have been obtained by assuming Gaussian input distributions. While optimal in single-user Gaussian channels, the issue with this assumption is that the Gaussian noise becomes the worst noise when the input distribution is also Gaussian. In this thesis, we propose a class of discrete constellations. We show that this class of constellations can automatically achieve the same sum rates as schemes that treat interference as noise or perform time sharing.

Standard statistical decision-making tools, such as inference, confidence intervals and forecasting, are contingent on the assumption that the statistical model used in the analysis is the true model. In linear mixed-effect models, ignoring model uncertainty results in an underestimation of the residual variance, contributing to hypothesis tests that demonstrate larger than nominal Type-I errors and confidence intervals with smaller than nominal coverage probabilities. A novel utilization of the generalized degrees of freedom developed by Zhang et al. (2012) is used to adjust the estimate of the residual variance for model uncertainty. Additionally, the general global linear approximation is extended to linear mixed-effect models to adjust the standard errors of the parameter estimates for model uncertainty. Both of these methods use a perturbation method for estimation, where random noise is added to the response variable and, conditional on the observed responses, the corresponding estimate is calculated. A simulation study demonstrates that when the proposed methodologies are utilized, both the variance and standard errors are inflated for model uncertainty. However, when a data-driven strategy is employed, the proposed methodologies show limited usefulness. These methods are evaluated with a trial assessing the performance of cervical traction in the treatment of cervical radiculopathy.