I would say the easiest way to determine this is to look as the classes as analogues of the PHB classes.
Alchemist: A spellcasting class (specifically a Wizard), so using the Wizard starting ages seems appropriate.
Cavalier: A martial class somewhere in-between the Paladin and Fighter; Fighter starting ages.
Inquisitor: A tricky one, as arguments can be made that it is an analog to Paladins, Rangers, or even Rogues. Nonetheless, I'd give them the Cleric starting age, to represent indoctrination of religious dogma from their Church.
Oracle: The Divine equivalent of a Sorcerer, so I think the Sorcerer starting ages works fine.
Summoner: Another Wizard for all intents and purposes, so again, the Wizard starting ages seems to work well.
Witch: The Witch strikes me as am Arcane Druid, so the Druid starting ages seems to suit.