**Title:** Making sense of noisy data: theory and applications

**Abstract**

This thesis introduces a novel and interpretable index of increase which is mathematically defined based on the distance between a given function and a set of non-increasing functions. Unlike the widely used traditional statistical methods for analyzing relationships between variables, the index does not rely on assumptions such as linearity, normality, and monotonicity, which may not be satisfied. Hence, it has the flexibility to be applied directly on pairs of data points to measure and compare non-linear, asymmetric, and non-monotonic relationships between two variables.

We begin with a review of the literature and background knowledge in Chapter 2.

In Chapter 3, we propose a distance-based index of increase, describe its properties in detail, and show its benefits through applying it to an educational dataset. In this way, we see the interpretability of the index of increase and how it can be applied. We also propose several modifications for di↵erent scenarios, such as subgroup analysis. Lastly, we provide a step-by-step implementation guideline for non-statistical researchers or practitioners.

In Chapter 4, we investigate two extensions of the index of increase, which quantify the interchangeability between variables. We discuss the usage of them in the context of developing curricula, accompanied with extensive graphical and numerical illustrations.

In Chapter 5, we introduce and explore an empirical index of increase that works in both deterministic and random environments, thus allowing to assess monotonicity of functions that are prone to random measurement-errors. We prove consistency of the empirical index and show how its rate of convergence is influenced by deterministic and random parts of the data. In particular, the obtained results suggest a frequency at which observations should be taken in order to reach any pre-specified level of estimation precision. We illustrate the index using data arising from purely deterministic and error-contaminated functions, which may or may not be monotonic.

Finally, in Chapter 6, we summarize our main results and give an outline of potential future works.

**Keywords:** index of increase; monotonicity; consistent estimator; distance-based measure; curriculum development; students performance evaluation