In forensic science, the strength of evidence is calculated mainly by statistical models called likelihood ratio systems. In court cases, the specific-source likelihood ratio system is used by forensic scientists to determine if a trace originates from a known reference, called the trace-reference problem. However, collecting sufficient data to create a specific source model may be time-consuming and costly. If the number of court cases becomes too high this could be problematic. Therefore there is a need for other models that can perform as well as a specific-source model if it is infeasible.

A common-source model could be a solution, as this model can be re-used over cases. To this end, we introduce two common-source systems: a common-source feature-based system and a common-source score-based system. We compare their performance to a specific-source score-based system in a trace-reference setting. The simulations show that the common source feature-based method is the best-performing likelihood ratio system if the dimensionality is not too high, and the sources are equally variable. The analysis shows that the common-source score-based method can work as effectively as a specific-source score-based model in certain scenarios.

Additionally, we researched a preprocessor, known as percentile rank, which aims to consider typicality for score-based methods. For the common-source score-based system, using a percentile-rank preprocessor can improve the performance for large sample sizes, while considering the rarity of the measurements.

In this thesis we shall consider sample covariance matrices Sn in the case when the dimension of the data increases with the sample size to infinity ,while the ratio approaches a fixed constant. We will derive a new statistic based on the general linear shrinkage estimator by Bodnar et al. (2014)[1] We will show that the new statistic is normally distributed under the null hypothesis that the true covariance matrix is the identity, where we assume the existence of the fourth moment of our data.

Furthermore, we will do simulation study that compares our new statistic to tests from finite dimensional statistics that have been altered to work in high dimensional statistics by Wang and Yao [3]. We will look at three different hypothesis, the equicorrelation case, the auto-regressive case and a fixed ratio case. After that, we will look at the non-linear shrinkage estimator based on the work by Ledoit and Peche (2011) [11], and show that, under the null hypothesis, constructing a test is not directly possible like it is in the linear case.