This thesis would never have been finished, if there would not exist many amazing people around me who helped, supported and motivated me during the last years. I really want to thank all who have contributed making that happen.First of all, I thank my first advisor Stephan Huckemann for his patience with me, for time for discussions, giving me the opportunity to travel and do research abroad and introducing me to the topic of non-Euclidean statistics with this exciting application of gait analysis. Further, I thank him for his careful proofreading of this thesis helping to improve its structure and eliminating a lot of bad habits in my writings. Secondly, I thank Michael Pierrynowski who recorded the data underlying this thesis, provided a lot of biomechanical insights and interesting articles. Especially, I want to thank him for the detailed explanations of the experimental setup and discussions about the data analysis. Real data analysis is always a great challenge and can be extremely frustrating from time to time, but nevertheless I am glad that I had the opportunity to study such a complex data set and had help from a biomechanics expert.I owe many thanks to Max Sommerfeld. I am not only grateful for his suggestions where I could look for answers, but mostly for his faith in me, his interest in my work, and for encouraging me to not give up. Without his support -sometimes only by drinking an espresso together and talking during a work break -I am not sure whether I would have finished my dissertation. Additionally, I thank him for proofreading Chapter 1 and providing a lot of constructive suggestions on making the presentation clearer.I also really want to thank my friends and colleagues Benjamin Heuer and Alexander Hartmann. They were often a great help in solving small mathematical puzzles and they resolved many of my confusions often by simply arguing that my question is simple. Among other useful inputs, I thank Benjamin Heuer, especially, for his discussions and explanations on uniform convergence of stochastic processes. However, I am also glad that our office hours did not had much of an overlap, since otherwise probably both of us would never have finished their dissertations, since there exist so many fun topics -other than mathematics -to talk about. To Alexander, I owe thanks for proofreading the introduction and providing very helpful Latex support. Last but not least, I thank both of them for really enjoyable lunch breaks, for making the office a fun place to work at and for entertaining board game evenings.I would like to thank Florian Boßmann from the numerics department for his vi Acknowledgments help in the development of the algorithm to solve the minimizing problem, which arose in the time warping procedure for curves in Lie groups. My second advisor, Axel Munk, I thank for helping me a few times with discussions, inspiring me to learn about simultaneous confidence bands and providing many interesting articles. I also thank Christian Böhm, the system administrator of the FBMS, for his com...