There is not much real hard data publicly available that shows benefits of using agile software development approaches. The data that is available doesn't bring you much either; what use is a statement that company X had a productivity increase after introducing Scrum? Who knows if the circumstances are comparable with yours? Who knows what other factors contributed to the sudden increase?
Running the pilot projects is, in my opinion, the way to proof its benefits. However, before plunging into the water, consider what your current problems are; try to express them in metrics like productivity, number of bugs found, time to market, user acceptance. Define the root causes for your most prominent problems, explain how agile would solve them, and after the pilot compare historic metrics with the new ones. That is what you need to convince management.
Oversimplified example: the time between software finished and user acceptance is too long. Cause: users see the software late in the process. Agile solution: close and early user involvement. You can measure the time it takes between completion and the formal acceptance. Should be better.
Of course, this is a very simple example, but keep in mind: every technique tries to solve a problem; if you don't have that problem, introducing the technique will not improve anything. Define the problem, define the metric, explain and implement the correct technique or method, and compare the metrics.More information: