Suppose that we want to infer the effect of a treatment on a certain outcome, where both the treatment and outcome are influenced by other variables. It has been well-established that in the linear setting, in case we know beforehand which of these other variables are instrumental (for the effect of the treatment on the outcome), we can infer the treatment effect in a consis-tent sense. This thesis analyses 3 methods that deals with the issue of unknown instrumental variables (IVs) and functional relationships in different ways to infer the treatment effect. The first method, Causal Inference with Invalid Instruments (CIII), assumes that we have a linear setting and a set with potential instrumental variables for whom a majority or plurality rule holds to obtain a robust confidence interval for the treatment effect. The second method, Anchor Regression (AR), only assumes a linear setting. By mediating between different meth-ods, the AR-estimator turns out the be robust to changes in the distribution of the sampled data. Lastly, Two Stage Curvature Identification (TSCI), does not require a linear setting or information on the IVs. Instead, it relies on the difference in functional form between the effect of the variables on the treatment and the effect of the variables on the outcome for consistent estimation and asymptotic normality. TSCI also provides a test for IV presence in the non-linear setting. In this thesis, I will explain the workings of these 3 methods, analyse their theoretical foundation and do simulation studies. Based on these analyses, I make several additions and suggestions to expand the theoretical scope and improve practical efficacy.3

In probability theory, Lp spaces for p > 0 together with the topology of conver-
gence in probability have been widely applied. However, in that case we restrict
ourselves to only a part of all the measurable functions and to an underlying prob-
ability space. One of the main aims of this thesis is to generalize this concept to
the set of all measurable functions with the usual a.e. equivalence classes (which
we call L0) and (possibly) non-finite measure spaces. The other main aim is to
establish an ordered structure on this L0 space