Crown chances for investigations are based on reward box:
1 Silver and 1 Gold - 6% (small) 3% (giant)
3 Gold and 1 Bronze - 6% (small) 3% (giant)
4 or more Gold - 6% (small) 6% (giant)
3 Purple (Tempered Investigations) - 6% (small) 3% (giant)
4 or more Purple (Tempered Investigations) - 6% (small) 6% (giant)
Pick the investigation that best suits your target size. For certain rare monsters like the Gold Rathian, it can be difficult to aquire suitable investigations. In this situation the arena quest is the best option. Whilst the crown chance is lower, Arena loads quickly even better if you have a solid state drive.
Like others mentioned if you post the crowns you need we can point you in the right direction.